KILLED
proof of input_LOCxx5HRyE.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) InliningProof [UPPER BOUND(ID), 139 ms]
        (16) CpxRNTS
        (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 85 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 56 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 121 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 41 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 13 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 89 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
        (44) CpxRNTS
        (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 1065 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 414 ms]
        (50) CpxRNTS
        (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (52) CpxRNTS
        (53) IntTrsBoundProof [UPPER BOUND(ID), 181 ms]
        (54) CpxRNTS
        (55) IntTrsBoundProof [UPPER BOUND(ID), 85 ms]
        (56) CpxRNTS
        (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (58) CpxRNTS
        (59) IntTrsBoundProof [UPPER BOUND(ID), 1969 ms]
        (60) CpxRNTS
        (61) IntTrsBoundProof [UPPER BOUND(ID), 3888 ms]
        (62) CpxRNTS
    (63) CompletionProof [UPPER BOUND(ID), 0 ms]
    (64) CpxTypedWeightedCompleteTrs
        (65) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (66) CpxRNTS
(67) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(68) CdtProblem
    (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 87 ms]
    (92) CdtProblem
    (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 91 ms]
    (96) CdtProblem
    (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms]
    (98) CdtProblem
    (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms]
    (120) CdtProblem
    (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (140) CdtProblem
    (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 380 ms]
    (154) CdtProblem
    (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 390 ms]
    (162) CdtProblem
    (163) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 396 ms]
    (164) CdtProblem
    (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem
    (175) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (176) CdtProblem
    (177) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 316 ms]
    (178) CdtProblem
    (179) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (180) CdtProblem
    (181) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (182) CdtProblem
    (183) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (184) CdtProblem
    (185) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (186) CdtProblem
    (187) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (188) CdtProblem
    (189) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (190) CdtProblem
    (191) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (192) CdtProblem
    (193) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (194) CdtProblem
    (195) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (196) CdtProblem
    (197) CdtLeafRemovalProof [ComplexityIfPolyImplication, 3 ms]
    (198) CdtProblem
    (199) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 455 ms]
    (200) CdtProblem
    (201) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (202) CdtProblem
    (203) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (204) CdtProblem
    (205) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (206) CdtProblem
    (207) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (208) CdtProblem
    (209) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (210) 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:

   empty(nil) -> true
   empty(cons(x, y)) -> false
   tail(nil) -> nil
   tail(cons(x, y)) -> y
   head(cons(x, y)) -> x
   zero(0) -> true
   zero(s(x)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   intlist(x) -> if_intlist(empty(x), x)
   if_intlist(true, x) -> nil
   if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x)))
   int(x, y) -> if_int(zero(x), zero(y), x, y)
   if_int(true, b, x, y) -> if1(b, x, y)
   if_int(false, b, x, y) -> if2(b, x, y)
   if1(true, x, y) -> cons(0, nil)
   if1(false, x, y) -> cons(0, int(s(0), y))
   if2(true, x, y) -> nil
   if2(false, x, y) -> intlist(int(p(x), p(y)))

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:

   empty(nil) -> true
   empty(cons(x, y)) -> false
   tail(nil) -> nil
   tail(cons(x, y)) -> y
   head(cons(x, y)) -> x
   zero(0') -> true
   zero(s(x)) -> false
   p(0') -> 0'
   p(s(0')) -> 0'
   p(s(s(x))) -> s(p(s(x)))
   intlist(x) -> if_intlist(empty(x), x)
   if_intlist(true, x) -> nil
   if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x)))
   int(x, y) -> if_int(zero(x), zero(y), x, y)
   if_int(true, b, x, y) -> if1(b, x, y)
   if_int(false, b, x, y) -> if2(b, x, y)
   if1(true, x, y) -> cons(0', nil)
   if1(false, x, y) -> cons(0', int(s(0'), y))
   if2(true, x, y) -> nil
   if2(false, x, y) -> intlist(int(p(x), p(y)))

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:

   empty(nil) -> true
   empty(cons(x, y)) -> false
   tail(nil) -> nil
   tail(cons(x, y)) -> y
   head(cons(x, y)) -> x
   zero(0) -> true
   zero(s(x)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   intlist(x) -> if_intlist(empty(x), x)
   if_intlist(true, x) -> nil
   if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x)))
   int(x, y) -> if_int(zero(x), zero(y), x, y)
   if_int(true, b, x, y) -> if1(b, x, y)
   if_int(false, b, x, y) -> if2(b, x, y)
   if1(true, x, y) -> cons(0, nil)
   if1(false, x, y) -> cons(0, int(s(0), y))
   if2(true, x, y) -> nil
   if2(false, x, y) -> intlist(int(p(x), p(y)))

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:

   empty(nil) -> true [1]
   empty(cons(x, y)) -> false [1]
   tail(nil) -> nil [1]
   tail(cons(x, y)) -> y [1]
   head(cons(x, y)) -> x [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   p(0) -> 0 [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   intlist(x) -> if_intlist(empty(x), x) [1]
   if_intlist(true, x) -> nil [1]
   if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) [1]
   int(x, y) -> if_int(zero(x), zero(y), x, y) [1]
   if_int(true, b, x, y) -> if1(b, x, y) [1]
   if_int(false, b, x, y) -> if2(b, x, y) [1]
   if1(true, x, y) -> cons(0, nil) [1]
   if1(false, x, y) -> cons(0, int(s(0), y)) [1]
   if2(true, x, y) -> nil [1]
   if2(false, x, y) -> intlist(int(p(x), p(y))) [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:

   empty(nil) -> true [1]
   empty(cons(x, y)) -> false [1]
   tail(nil) -> nil [1]
   tail(cons(x, y)) -> y [1]
   head(cons(x, y)) -> x [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   p(0) -> 0 [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   intlist(x) -> if_intlist(empty(x), x) [1]
   if_intlist(true, x) -> nil [1]
   if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) [1]
   int(x, y) -> if_int(zero(x), zero(y), x, y) [1]
   if_int(true, b, x, y) -> if1(b, x, y) [1]
   if_int(false, b, x, y) -> if2(b, x, y) [1]
   if1(true, x, y) -> cons(0, nil) [1]
   if1(false, x, y) -> cons(0, int(s(0), y)) [1]
   if2(true, x, y) -> nil [1]
   if2(false, x, y) -> intlist(int(p(x), p(y))) [1]

The TRS has the following type information:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: 0:s -> nil:cons -> nil:cons
false :: true:false
tail :: nil:cons -> nil:cons
head :: nil:cons -> 0:s
zero :: 0:s -> true:false
0 :: 0:s
s :: 0:s -> 0:s
p :: 0:s -> 0:s
intlist :: nil:cons -> nil:cons
if_intlist :: true:false -> nil:cons -> nil:cons
int :: 0:s -> 0:s -> nil:cons
if_int :: true:false -> true:false -> 0:s -> 0:s -> nil:cons
if1 :: true:false -> 0:s -> 0:s -> nil:cons
if2 :: true:false -> 0:s -> 0:s -> nil:cons

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:

   tail_1
   zero_1
   int_2
   p_1
   empty_1
   if_int_4
   if2_3
   if1_3
   intlist_1
   if_intlist_2
   head_1

Due to the following rules being added:

   head(v0) -> 0 [0]

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:

   empty(nil) -> true [1]
   empty(cons(x, y)) -> false [1]
   tail(nil) -> nil [1]
   tail(cons(x, y)) -> y [1]
   head(cons(x, y)) -> x [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   p(0) -> 0 [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   intlist(x) -> if_intlist(empty(x), x) [1]
   if_intlist(true, x) -> nil [1]
   if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) [1]
   int(x, y) -> if_int(zero(x), zero(y), x, y) [1]
   if_int(true, b, x, y) -> if1(b, x, y) [1]
   if_int(false, b, x, y) -> if2(b, x, y) [1]
   if1(true, x, y) -> cons(0, nil) [1]
   if1(false, x, y) -> cons(0, int(s(0), y)) [1]
   if2(true, x, y) -> nil [1]
   if2(false, x, y) -> intlist(int(p(x), p(y))) [1]
   head(v0) -> 0 [0]

The TRS has the following type information:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: 0:s -> nil:cons -> nil:cons
false :: true:false
tail :: nil:cons -> nil:cons
head :: nil:cons -> 0:s
zero :: 0:s -> true:false
0 :: 0:s
s :: 0:s -> 0:s
p :: 0:s -> 0:s
intlist :: nil:cons -> nil:cons
if_intlist :: true:false -> nil:cons -> nil:cons
int :: 0:s -> 0:s -> nil:cons
if_int :: true:false -> true:false -> 0:s -> 0:s -> nil:cons
if1 :: true:false -> 0:s -> 0:s -> nil:cons
if2 :: true:false -> 0:s -> 0:s -> nil:cons

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:

   empty(nil) -> true [1]
   empty(cons(x, y)) -> false [1]
   tail(nil) -> nil [1]
   tail(cons(x, y)) -> y [1]
   head(cons(x, y)) -> x [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   p(0) -> 0 [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   intlist(nil) -> if_intlist(true, nil) [2]
   intlist(cons(x', y')) -> if_intlist(false, cons(x', y')) [2]
   if_intlist(true, x) -> nil [1]
   if_intlist(false, nil) -> cons(s(head(nil)), intlist(nil)) [2]
   if_intlist(false, cons(x'', y'')) -> cons(s(head(cons(x'', y''))), intlist(y'')) [2]
   int(0, 0) -> if_int(true, true, 0, 0) [3]
   int(0, s(x2)) -> if_int(true, false, 0, s(x2)) [3]
   int(s(x1), 0) -> if_int(false, true, s(x1), 0) [3]
   int(s(x1), s(x3)) -> if_int(false, false, s(x1), s(x3)) [3]
   if_int(true, b, x, y) -> if1(b, x, y) [1]
   if_int(false, b, x, y) -> if2(b, x, y) [1]
   if1(true, x, y) -> cons(0, nil) [1]
   if1(false, x, y) -> cons(0, int(s(0), y)) [1]
   if2(true, x, y) -> nil [1]
   if2(false, 0, 0) -> intlist(int(0, 0)) [3]
   if2(false, 0, s(0)) -> intlist(int(0, 0)) [3]
   if2(false, 0, s(s(x5))) -> intlist(int(0, s(p(s(x5))))) [3]
   if2(false, s(0), 0) -> intlist(int(0, 0)) [3]
   if2(false, s(0), s(0)) -> intlist(int(0, 0)) [3]
   if2(false, s(0), s(s(x6))) -> intlist(int(0, s(p(s(x6))))) [3]
   if2(false, s(s(x4)), 0) -> intlist(int(s(p(s(x4))), 0)) [3]
   if2(false, s(s(x4)), s(0)) -> intlist(int(s(p(s(x4))), 0)) [3]
   if2(false, s(s(x4)), s(s(x7))) -> intlist(int(s(p(s(x4))), s(p(s(x7))))) [3]
   head(v0) -> 0 [0]

The TRS has the following type information:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: 0:s -> nil:cons -> nil:cons
false :: true:false
tail :: nil:cons -> nil:cons
head :: nil:cons -> 0:s
zero :: 0:s -> true:false
0 :: 0:s
s :: 0:s -> 0:s
p :: 0:s -> 0:s
intlist :: nil:cons -> nil:cons
if_intlist :: true:false -> nil:cons -> nil:cons
int :: 0:s -> 0:s -> nil:cons
if_int :: true:false -> true:false -> 0:s -> 0:s -> nil:cons
if1 :: true:false -> 0:s -> 0:s -> nil:cons
if2 :: true:false -> 0:s -> 0:s -> nil:cons

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:

   nil => 0
   true => 1
   false => 0
   0 => 0

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

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

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + x5))) :|: x5 >= 0, z'' = 1 + (1 + x5), z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + x6))) :|: z'' = 1 + (1 + x6), x6 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + x4), 0)) :|: z'' = 0, z' = 1 + (1 + x4), x4 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + x4), 0)) :|: z' = 1 + (1 + x4), x4 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + x4), 1 + p(1 + x7))) :|: z' = 1 + (1 + x4), x4 >= 0, z'' = 1 + (1 + x7), x7 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(b, x, y) :|: b >= 0, z1 = y, x >= 0, y >= 0, z' = b, z'' = x, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(b, x, y) :|: b >= 0, z1 = y, z = 1, x >= 0, y >= 0, z' = b, z'' = x
   if_intlist(z, z') -{ 1 }-> 0 :|: z' = x, z = 1, x >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + head(0)) + intlist(0) :|: z = 0, z' = 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + head(1 + x'' + y'')) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + x2) :|: z' = 1 + x2, z = 0, x2 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + x1, 0) :|: x1 >= 0, z = 1 + x1, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + x1, 1 + x3) :|: x1 >= 0, z' = 1 + x3, z = 1 + x1, x3 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z = 1 + (1 + x)
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x


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

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

   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

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

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

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + x5))) :|: x5 >= 0, z'' = 1 + (1 + x5), z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + x6))) :|: z'' = 1 + (1 + x6), x6 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + x4), 0)) :|: z'' = 0, z' = 1 + (1 + x4), x4 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + x4), 0)) :|: z' = 1 + (1 + x4), x4 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + x4), 1 + p(1 + x7))) :|: z' = 1 + (1 + x4), x4 >= 0, z'' = 1 + (1 + x7), x7 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(b, x, y) :|: b >= 0, z1 = y, x >= 0, y >= 0, z' = b, z'' = x, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(b, x, y) :|: b >= 0, z1 = y, z = 1, x >= 0, y >= 0, z' = b, z'' = x
   if_intlist(z, z') -{ 1 }-> 0 :|: z' = x, z = 1, x >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + x2) :|: z' = 1 + x2, z = 0, x2 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + x1, 0) :|: x1 >= 0, z = 1 + x1, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + x1, 1 + x3) :|: x1 >= 0, z' = 1 + x3, z = 1 + x1, x3 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z = 1 + (1 + x)
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x


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

(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:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0


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

(19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { empty }
   { zero }
   { tail }
   { head }
   { if_intlist, intlist }
   { p }
   { if_int, int, if2, if1 }

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

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {empty}, {zero}, {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1}

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

(21) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

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

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {empty}, {zero}, {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: empty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {empty}, {zero}, {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: ?, size: O(1) [1]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: empty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {zero}, {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]

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

(27) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {zero}, {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: zero
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {zero}, {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: ?, size: O(1) [1]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: zero
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]

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

(33) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: tail
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: ?, size: O(n^1) [z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: tail
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]

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

(39) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: head
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  head: runtime: ?, size: O(n^1) [z]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: head
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {if_intlist,intlist}, {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  head: runtime: O(1) [1], size: O(n^1) [z]

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

(45) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {if_intlist,intlist}, {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  head: runtime: O(1) [1], size: O(n^1) [z]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: if_intlist
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 4 + 2*z' + z'^2

Computed SIZE bound using KoAT for: intlist
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 11 + 8*z + 4*z^2

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {if_intlist,intlist}, {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  head: runtime: O(1) [1], size: O(n^1) [z]
  if_intlist: runtime: ?, size: O(n^2) [4 + 2*z' + z'^2]
  intlist: runtime: ?, size: O(n^2) [11 + 8*z + 4*z^2]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: if_intlist
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 11 + 5*z'

Computed RUNTIME bound using CoFloCo for: intlist
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 13 + 5*z

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

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0
   intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  head: runtime: O(1) [1], size: O(n^1) [z]
  if_intlist: runtime: O(n^1) [11 + 5*z'], size: O(n^2) [4 + 2*z' + z'^2]
  intlist: runtime: O(n^1) [13 + 5*z], size: O(n^2) [11 + 8*z + 4*z^2]

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

(51) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 16 + 5*y'' }-> 1 + (1 + x) + s1 :|: s1 >= 0, s1 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 15 }-> 1 + (1 + 0) + s'' :|: s'' >= 0, s'' <= 8 * 0 + 4 * (0 * 0) + 11, z = 0, z' = 0, v0 >= 0, 0 = v0
   if_intlist(z, z') -{ 15 + 5*y'' }-> 1 + (1 + 0) + s2 :|: s2 >= 0, s2 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 13 }-> s :|: s >= 0, s <= 2 * 0 + 4 + 0 * 0, z = 0
   intlist(z) -{ 18 + 5*x' + 5*y' }-> s' :|: s' >= 0, s' <= 2 * (1 + x' + y') + 4 + (1 + x' + y') * (1 + x' + y'), x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  head: runtime: O(1) [1], size: O(n^1) [z]
  if_intlist: runtime: O(n^1) [11 + 5*z'], size: O(n^2) [4 + 2*z' + z'^2]
  intlist: runtime: O(n^1) [13 + 5*z], size: O(n^2) [11 + 8*z + 4*z^2]

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: p
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 16 + 5*y'' }-> 1 + (1 + x) + s1 :|: s1 >= 0, s1 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 15 }-> 1 + (1 + 0) + s'' :|: s'' >= 0, s'' <= 8 * 0 + 4 * (0 * 0) + 11, z = 0, z' = 0, v0 >= 0, 0 = v0
   if_intlist(z, z') -{ 15 + 5*y'' }-> 1 + (1 + 0) + s2 :|: s2 >= 0, s2 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 13 }-> s :|: s >= 0, s <= 2 * 0 + 4 + 0 * 0, z = 0
   intlist(z) -{ 18 + 5*x' + 5*y' }-> s' :|: s' >= 0, s' <= 2 * (1 + x' + y') + 4 + (1 + x' + y') * (1 + x' + y'), x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  head: runtime: O(1) [1], size: O(n^1) [z]
  if_intlist: runtime: O(n^1) [11 + 5*z'], size: O(n^2) [4 + 2*z' + z'^2]
  intlist: runtime: O(n^1) [13 + 5*z], size: O(n^2) [11 + 8*z + 4*z^2]
  p: runtime: ?, size: O(n^1) [z]

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: p
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z

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

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 16 + 5*y'' }-> 1 + (1 + x) + s1 :|: s1 >= 0, s1 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 15 }-> 1 + (1 + 0) + s'' :|: s'' >= 0, s'' <= 8 * 0 + 4 * (0 * 0) + 11, z = 0, z' = 0, v0 >= 0, 0 = v0
   if_intlist(z, z') -{ 15 + 5*y'' }-> 1 + (1 + 0) + s2 :|: s2 >= 0, s2 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 13 }-> s :|: s >= 0, s <= 2 * 0 + 4 + 0 * 0, z = 0
   intlist(z) -{ 18 + 5*x' + 5*y' }-> s' :|: s' >= 0, s' <= 2 * (1 + x' + y') + 4 + (1 + x' + y') * (1 + x' + y'), x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  head: runtime: O(1) [1], size: O(n^1) [z]
  if_intlist: runtime: O(n^1) [11 + 5*z'], size: O(n^2) [4 + 2*z' + z'^2]
  intlist: runtime: O(n^1) [13 + 5*z], size: O(n^2) [11 + 8*z + 4*z^2]
  p: runtime: O(n^1) [2 + z], size: O(n^1) [z]

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

(57) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 4 + z'' }-> intlist(int(0, 1 + s4)) :|: s4 >= 0, s4 <= 1 + (z'' - 2), z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 4 + z'' }-> intlist(int(0, 1 + s5)) :|: s5 >= 0, s5 <= 1 + (z'' - 2), z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 4 + z' }-> intlist(int(1 + s6, 0)) :|: s6 >= 0, s6 <= 1 + (z' - 2), z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 4 + z' }-> intlist(int(1 + s7, 0)) :|: s7 >= 0, s7 <= 1 + (z' - 2), z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 5 + z' + z'' }-> intlist(int(1 + s8, 1 + s9)) :|: s8 >= 0, s8 <= 1 + (z' - 2), s9 >= 0, s9 <= 1 + (z'' - 2), z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 16 + 5*y'' }-> 1 + (1 + x) + s1 :|: s1 >= 0, s1 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 15 }-> 1 + (1 + 0) + s'' :|: s'' >= 0, s'' <= 8 * 0 + 4 * (0 * 0) + 11, z = 0, z' = 0, v0 >= 0, 0 = v0
   if_intlist(z, z') -{ 15 + 5*y'' }-> 1 + (1 + 0) + s2 :|: s2 >= 0, s2 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 13 }-> s :|: s >= 0, s <= 2 * 0 + 4 + 0 * 0, z = 0
   intlist(z) -{ 18 + 5*x' + 5*y' }-> s' :|: s' >= 0, s' <= 2 * (1 + x' + y') + 4 + (1 + x' + y') * (1 + x' + y'), x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 2 + z }-> 1 + s3 :|: s3 >= 0, s3 <= 1 + (z - 2), z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  head: runtime: O(1) [1], size: O(n^1) [z]
  if_intlist: runtime: O(n^1) [11 + 5*z'], size: O(n^2) [4 + 2*z' + z'^2]
  intlist: runtime: O(n^1) [13 + 5*z], size: O(n^2) [11 + 8*z + 4*z^2]
  p: runtime: O(n^1) [2 + z], size: O(n^1) [z]

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: if2
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

Computed SIZE bound using CoFloCo for: if1
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 4 + z'' }-> intlist(int(0, 1 + s4)) :|: s4 >= 0, s4 <= 1 + (z'' - 2), z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 4 + z'' }-> intlist(int(0, 1 + s5)) :|: s5 >= 0, s5 <= 1 + (z'' - 2), z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 4 + z' }-> intlist(int(1 + s6, 0)) :|: s6 >= 0, s6 <= 1 + (z' - 2), z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 4 + z' }-> intlist(int(1 + s7, 0)) :|: s7 >= 0, s7 <= 1 + (z' - 2), z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 5 + z' + z'' }-> intlist(int(1 + s8, 1 + s9)) :|: s8 >= 0, s8 <= 1 + (z' - 2), s9 >= 0, s9 <= 1 + (z'' - 2), z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 16 + 5*y'' }-> 1 + (1 + x) + s1 :|: s1 >= 0, s1 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 15 }-> 1 + (1 + 0) + s'' :|: s'' >= 0, s'' <= 8 * 0 + 4 * (0 * 0) + 11, z = 0, z' = 0, v0 >= 0, 0 = v0
   if_intlist(z, z') -{ 15 + 5*y'' }-> 1 + (1 + 0) + s2 :|: s2 >= 0, s2 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 13 }-> s :|: s >= 0, s <= 2 * 0 + 4 + 0 * 0, z = 0
   intlist(z) -{ 18 + 5*x' + 5*y' }-> s' :|: s' >= 0, s' <= 2 * (1 + x' + y') + 4 + (1 + x' + y') * (1 + x' + y'), x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 2 + z }-> 1 + s3 :|: s3 >= 0, s3 <= 1 + (z - 2), z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  head: runtime: O(1) [1], size: O(n^1) [z]
  if_intlist: runtime: O(n^1) [11 + 5*z'], size: O(n^2) [4 + 2*z' + z'^2]
  intlist: runtime: O(n^1) [13 + 5*z], size: O(n^2) [11 + 8*z + 4*z^2]
  p: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  if_int: runtime: ?, size: INF
  int: runtime: ?, size: INF
  if2: runtime: ?, size: INF
  if1: runtime: ?, size: INF

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: if_int
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 4 + z'' }-> intlist(int(0, 1 + s4)) :|: s4 >= 0, s4 <= 1 + (z'' - 2), z'' - 2 >= 0, z = 0, z' = 0
   if2(z, z', z'') -{ 4 + z'' }-> intlist(int(0, 1 + s5)) :|: s5 >= 0, s5 <= 1 + (z'' - 2), z'' - 2 >= 0, z' = 1 + 0, z = 0
   if2(z, z', z'') -{ 4 + z' }-> intlist(int(1 + s6, 0)) :|: s6 >= 0, s6 <= 1 + (z' - 2), z'' = 0, z' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 4 + z' }-> intlist(int(1 + s7, 0)) :|: s7 >= 0, s7 <= 1 + (z' - 2), z' - 2 >= 0, z = 0, z'' = 1 + 0
   if2(z, z', z'') -{ 5 + z' + z'' }-> intlist(int(1 + s8, 1 + s9)) :|: s8 >= 0, s8 <= 1 + (z' - 2), s9 >= 0, s9 <= 1 + (z'' - 2), z' - 2 >= 0, z'' - 2 >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   if_intlist(z, z') -{ 16 + 5*y'' }-> 1 + (1 + x) + s1 :|: s1 >= 0, s1 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0
   if_intlist(z, z') -{ 15 }-> 1 + (1 + 0) + s'' :|: s'' >= 0, s'' <= 8 * 0 + 4 * (0 * 0) + 11, z = 0, z' = 0, v0 >= 0, 0 = v0
   if_intlist(z, z') -{ 15 + 5*y'' }-> 1 + (1 + 0) + s2 :|: s2 >= 0, s2 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0
   int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0
   int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   intlist(z) -{ 13 }-> s :|: s >= 0, s <= 2 * 0 + 4 + 0 * 0, z = 0
   intlist(z) -{ 18 + 5*x' + 5*y' }-> s' :|: s' >= 0, s' <= 2 * (1 + x' + y') + 4 + (1 + x' + y') * (1 + x' + y'), x' >= 0, y' >= 0, z = 1 + x' + y'
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 2 + z }-> 1 + s3 :|: s3 >= 0, s3 <= 1 + (z - 2), z - 2 >= 0
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: z - 1 >= 0

Function symbols to be analyzed: {if_int,int,if2,if1}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [1]
  zero: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  head: runtime: O(1) [1], size: O(n^1) [z]
  if_intlist: runtime: O(n^1) [11 + 5*z'], size: O(n^2) [4 + 2*z' + z'^2]
  intlist: runtime: O(n^1) [13 + 5*z], size: O(n^2) [11 + 8*z + 4*z^2]
  p: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  if_int: runtime: INF, size: INF
  int: runtime: ?, size: INF
  if2: runtime: ?, size: INF
  if1: runtime: ?, size: INF

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

(63) 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:

   head(v0) -> null_head [0]
   zero(v0) -> null_zero [0]
   p(v0) -> null_p [0]
   if_intlist(v0, v1) -> null_if_intlist [0]
   if_int(v0, v1, v2, v3) -> null_if_int [0]
   if1(v0, v1, v2) -> null_if1 [0]
   if2(v0, v1, v2) -> null_if2 [0]
   empty(v0) -> null_empty [0]
   tail(v0) -> null_tail [0]

And the following fresh constants:    null_head, null_zero, null_p, null_if_intlist, null_if_int, null_if1, null_if2, null_empty, null_tail

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

(64)
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:

   empty(nil) -> true [1]
   empty(cons(x, y)) -> false [1]
   tail(nil) -> nil [1]
   tail(cons(x, y)) -> y [1]
   head(cons(x, y)) -> x [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   p(0) -> 0 [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   intlist(x) -> if_intlist(empty(x), x) [1]
   if_intlist(true, x) -> nil [1]
   if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) [1]
   int(x, y) -> if_int(zero(x), zero(y), x, y) [1]
   if_int(true, b, x, y) -> if1(b, x, y) [1]
   if_int(false, b, x, y) -> if2(b, x, y) [1]
   if1(true, x, y) -> cons(0, nil) [1]
   if1(false, x, y) -> cons(0, int(s(0), y)) [1]
   if2(true, x, y) -> nil [1]
   if2(false, x, y) -> intlist(int(p(x), p(y))) [1]
   head(v0) -> null_head [0]
   zero(v0) -> null_zero [0]
   p(v0) -> null_p [0]
   if_intlist(v0, v1) -> null_if_intlist [0]
   if_int(v0, v1, v2, v3) -> null_if_int [0]
   if1(v0, v1, v2) -> null_if1 [0]
   if2(v0, v1, v2) -> null_if2 [0]
   empty(v0) -> null_empty [0]
   tail(v0) -> null_tail [0]

The TRS has the following type information:
empty :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail -> true:false:null_zero:null_empty
nil :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail
true :: true:false:null_zero:null_empty
cons :: 0:s:null_head:null_p -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail
false :: true:false:null_zero:null_empty
tail :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail
head :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail -> 0:s:null_head:null_p
zero :: 0:s:null_head:null_p -> true:false:null_zero:null_empty
0 :: 0:s:null_head:null_p
s :: 0:s:null_head:null_p -> 0:s:null_head:null_p
p :: 0:s:null_head:null_p -> 0:s:null_head:null_p
intlist :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail
if_intlist :: true:false:null_zero:null_empty -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail
int :: 0:s:null_head:null_p -> 0:s:null_head:null_p -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail
if_int :: true:false:null_zero:null_empty -> true:false:null_zero:null_empty -> 0:s:null_head:null_p -> 0:s:null_head:null_p -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail
if1 :: true:false:null_zero:null_empty -> 0:s:null_head:null_p -> 0:s:null_head:null_p -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail
if2 :: true:false:null_zero:null_empty -> 0:s:null_head:null_p -> 0:s:null_head:null_p -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail
null_head :: 0:s:null_head:null_p
null_zero :: true:false:null_zero:null_empty
null_p :: 0:s:null_head:null_p
null_if_intlist :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail
null_if_int :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail
null_if1 :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail
null_if2 :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail
null_empty :: true:false:null_zero:null_empty
null_tail :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail

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

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

   nil => 0
   true => 2
   false => 1
   0 => 0
   null_head => 0
   null_zero => 0
   null_p => 0
   null_if_intlist => 0
   null_if_int => 0
   null_if1 => 0
   null_if2 => 0
   null_empty => 0
   null_tail => 0

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

(66)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z) -{ 1 }-> 2 :|: z = 0
   empty(z) -{ 1 }-> 1 :|: z = 1 + x + y, x >= 0, y >= 0
   empty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   if1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0
   if2(z, z', z'') -{ 1 }-> intlist(int(p(x), p(y))) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0
   if2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_int(z, z', z'', z1) -{ 1 }-> if2(b, x, y) :|: b >= 0, z1 = y, z = 1, x >= 0, y >= 0, z' = b, z'' = x
   if_int(z, z', z'', z1) -{ 1 }-> if1(b, x, y) :|: z = 2, b >= 0, z1 = y, x >= 0, y >= 0, z' = b, z'' = x
   if_int(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   if_intlist(z, z') -{ 1 }-> 0 :|: z = 2, z' = x, x >= 0
   if_intlist(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   if_intlist(z, z') -{ 1 }-> 1 + (1 + head(x)) + intlist(tail(x)) :|: z' = x, z = 1, x >= 0
   int(z, z') -{ 1 }-> if_int(zero(x), zero(y), x, y) :|: x >= 0, y >= 0, z = x, z' = y
   intlist(z) -{ 1 }-> if_intlist(empty(x), x) :|: x >= 0, z = x
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z = 1 + (1 + x)
   tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   tail(z) -{ 1 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   zero(z) -{ 1 }-> 2 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: x >= 0, z = 1 + x
   zero(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   EMPTY(nil) -> c
   EMPTY(cons(z0, z1)) -> c1
   TAIL(nil) -> c2
   TAIL(cons(z0, z1)) -> c3
   HEAD(cons(z0, z1)) -> c4
   ZERO(0) -> c5
   ZERO(s(z0)) -> c6
   P(0) -> c7
   P(s(0)) -> c8
   P(s(s(z0))) -> c9(P(s(z0)))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
   IF_INTLIST(true, z0) -> c11
   IF_INTLIST(false, z0) -> c12(HEAD(z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(true, z0, z1) -> c18
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(true, z0, z1) -> c20
   IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
S tuples:
   EMPTY(nil) -> c
   EMPTY(cons(z0, z1)) -> c1
   TAIL(nil) -> c2
   TAIL(cons(z0, z1)) -> c3
   HEAD(cons(z0, z1)) -> c4
   ZERO(0) -> c5
   ZERO(s(z0)) -> c6
   P(0) -> c7
   P(s(0)) -> c8
   P(s(s(z0))) -> c9(P(s(z0)))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
   IF_INTLIST(true, z0) -> c11
   IF_INTLIST(false, z0) -> c12(HEAD(z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(true, z0, z1) -> c18
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(true, z0, z1) -> c20
   IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
K tuples:none
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   EMPTY_1, TAIL_1, HEAD_1, ZERO_1, P_1, INTLIST_1, IF_INTLIST_2, INT_2, IF_INT_4, IF1_3, IF2_3

Compound Symbols:   c, c1, c2, c3, c4, c5, c6, c7, c8, c9_1, c10_2, c11, c12_1, c13_2, c14_2, c15_2, c16_1, c17_1, c18, c19_1, c20, c21_3, c22_3


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

(69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 13 trailing nodes:
   IF2(true, z0, z1) -> c20
   IF_INTLIST(true, z0) -> c11
   P(0) -> c7
   HEAD(cons(z0, z1)) -> c4
   ZERO(0) -> c5
   TAIL(cons(z0, z1)) -> c3
   IF1(true, z0, z1) -> c18
   EMPTY(cons(z0, z1)) -> c1
   TAIL(nil) -> c2
   P(s(0)) -> c8
   ZERO(s(z0)) -> c6
   IF_INTLIST(false, z0) -> c12(HEAD(z0))
   EMPTY(nil) -> c

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
K tuples:none
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, INTLIST_1, IF_INTLIST_2, INT_2, IF_INT_4, IF1_3, IF2_3

Compound Symbols:   c9_1, c10_2, c13_2, c14_2, c15_2, c16_1, c17_1, c19_1, c21_3, c22_3


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

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
K tuples:none
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c21_3, c22_3, c10_1, c13_1, c14_1, c15_1


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

(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0)) by 
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(0))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(s(0)))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(0))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(s(0)))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(0))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(s(0)))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
K tuples:none
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c22_3, c10_1, c13_1, c14_1, c15_1, c21_3


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

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
K tuples:none
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c22_3, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1)) by 
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(0))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(s(0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(0))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(s(0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(0))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(s(0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
K tuples:none
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c22_3


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

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
K tuples:none
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c22_3, c22_2


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

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0)) by 
   INTLIST(nil) -> c10(IF_INTLIST(true, nil))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(nil) -> c10(IF_INTLIST(true, nil))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(nil) -> c10(IF_INTLIST(true, nil))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF_INTLIST_2, INT_2, IF2_3, INTLIST_1

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c13_1, c14_1, c15_1, c21_3, c21_2, c22_3, c22_2, c10_1


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

(83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   INTLIST(nil) -> c10(IF_INTLIST(true, nil))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF_INTLIST_2, INT_2, IF2_3, INTLIST_1

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c13_1, c14_1, c15_1, c21_3, c21_2, c22_3, c22_2, c10_1


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

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) by 
   IF_INTLIST(false, nil) -> c13(INTLIST(nil))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, nil) -> c13(INTLIST(nil))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, nil) -> c13(INTLIST(nil))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
K tuples:none
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INT_2, IF2_3, INTLIST_1, IF_INTLIST_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c14_1, c15_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1


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

(87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IF_INTLIST(false, nil) -> c13(INTLIST(nil))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
K tuples:none
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INT_2, IF2_3, INTLIST_1, IF_INTLIST_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c14_1, c15_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1


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

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) by 
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
K tuples:none
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INT_2, IF2_3, INTLIST_1, IF_INTLIST_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c15_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1


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

(91) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
We considered the (Usable) Rules:
   zero(0) -> true
   zero(s(z0)) -> false
And the Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IF1(x_1, x_2, x_3)) = x_1
   POL(IF2(x_1, x_2, x_3)) = x_1
   POL(IF_INT(x_1, x_2, x_3, x_4)) = x_2
   POL(IF_INTLIST(x_1, x_2)) = 0
   POL(INT(x_1, x_2)) = [1]
   POL(INTLIST(x_1)) = 0
   POL(P(x_1)) = 0
   POL(c10(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1
   POL(empty(x_1)) = [1] + x_1
   POL(false) = [1]
   POL(head(x_1)) = [1] + x_1
   POL(if1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if2(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if_int(x_1, x_2, x_3, x_4)) = [1]
   POL(if_intlist(x_1, x_2)) = [1] + x_1 + x_2
   POL(int(x_1, x_2)) = [1]
   POL(intlist(x_1)) = [1]
   POL(nil) = [1]
   POL(p(x_1)) = 0
   POL(s(x_1)) = 0
   POL(tail(x_1)) = [1] + x_1
   POL(true) = 0
   POL(zero(x_1)) = [1]

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INT_2, IF2_3, INTLIST_1, IF_INTLIST_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c15_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1


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

(93) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) by 
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1


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

(95) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
We considered the (Usable) Rules:
   zero(0) -> true
   zero(s(z0)) -> false
And the Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IF1(x_1, x_2, x_3)) = x_1
   POL(IF2(x_1, x_2, x_3)) = x_1
   POL(IF_INT(x_1, x_2, x_3, x_4)) = x_2
   POL(IF_INTLIST(x_1, x_2)) = 0
   POL(INT(x_1, x_2)) = [1]
   POL(INTLIST(x_1)) = 0
   POL(P(x_1)) = 0
   POL(c10(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1
   POL(empty(x_1)) = [1] + x_1
   POL(false) = [1]
   POL(head(x_1)) = [1] + x_1
   POL(if1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if2(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if_int(x_1, x_2, x_3, x_4)) = [1]
   POL(if_intlist(x_1, x_2)) = [1] + x_1 + x_2
   POL(int(x_1, x_2)) = [1]
   POL(intlist(x_1)) = [1]
   POL(nil) = [1]
   POL(p(x_1)) = 0
   POL(s(x_1)) = 0
   POL(tail(x_1)) = [1] + x_1
   POL(true) = 0
   POL(zero(x_1)) = [1]

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1


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

(97) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) by 
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(0))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(s(0)))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(0))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(s(0)))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(0))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(s(0)))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(0))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(s(0)))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(0))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(s(0)))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(0))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(s(0)))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1


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

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1


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

(101) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) by 
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)), P(0))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)), P(s(0)))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)), P(0))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)), P(s(0)))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)), P(0))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)), P(s(0)))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_1


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

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_1


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

(105) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) by 
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))), P(0))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))), P(s(0)))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))), P(0))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))), P(s(0)))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))), P(0))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))), P(s(0)))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_1


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

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_1


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

(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) by 
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(0))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(0)))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(0))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(0)))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(0))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(0)))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_1


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

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_1


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

(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) by 
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(x0)), s(0)) -> c21(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))), P(s(s(x0))))
   IF2(false, s(s(x0)), s(s(z0))) -> c21(INTLIST(int(s(p(s(x0))), s(p(s(z0))))), INT(p(s(s(x0))), p(s(s(z0)))), P(s(s(x0))))
   IF2(false, s(s(x0)), 0) -> c21(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1


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

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) by 
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(z0))) -> c21(INTLIST(int(0, s(p(s(z0))))), INT(p(0), p(s(s(z0)))))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1


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

(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) by 
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(0), s(s(z0))) -> c21(INTLIST(int(0, s(p(s(z0))))), INT(p(s(0)), p(s(s(z0)))))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1


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

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) by 
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(0))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(s(0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(0))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(s(0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(0))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(s(0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(0))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(s(0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(0))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(s(0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(0))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(s(0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1


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

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1


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

(123) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) by 
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1


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

(125) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) by 
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)), P(0))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))), P(s(0)))
   IF2(false, 0, s(s(z0))) -> c22(INTLIST(int(0, s(p(s(z0))))), INT(p(0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)), P(0))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))), P(s(0)))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)), P(0))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))), P(s(0)))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_1


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

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_1


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

(129) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) by 
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)), P(0))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))), P(s(0)))
   IF2(false, s(0), s(s(z0))) -> c22(INTLIST(int(0, s(p(s(z0))))), INT(p(s(0)), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)), P(0))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))), P(s(0)))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)), P(0))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))), P(s(0)))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_1


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

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_1


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

(133) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) by 
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))), P(s(0)))
   IF2(false, s(s(x0)), s(s(z0))) -> c22(INTLIST(int(s(p(s(x0))), s(p(s(z0))))), INT(p(s(s(x0))), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)), P(0))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))), P(s(0)))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)), P(0))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))), P(s(0)))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)), P(0))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_1


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

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_1


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

(137) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) by 
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(s(z0)), 0) -> c22(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_1


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(139) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) by 
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(z0)), s(0)) -> c22(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))

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(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1


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

(141) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) by 
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1


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

(143) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) by 
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1


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

(145) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) by 
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0)))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1


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

(147) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) by 
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1


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

(149) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) by IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
----------------------------------------

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1


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

(151) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) by IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
----------------------------------------

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1


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

(153) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
We considered the (Usable) Rules:
   p(0) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
And the Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(IF1(x_1, x_2, x_3)) = x_3
   POL(IF2(x_1, x_2, x_3)) = x_3
   POL(IF_INT(x_1, x_2, x_3, x_4)) = x_1 + x_4
   POL(IF_INTLIST(x_1, x_2)) = x_1
   POL(INT(x_1, x_2)) = x_2
   POL(INTLIST(x_1)) = 0
   POL(P(x_1)) = 0
   POL(c10(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c22(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1
   POL(empty(x_1)) = x_1
   POL(false) = 0
   POL(head(x_1)) = [1] + x_1
   POL(if1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if2(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if_int(x_1, x_2, x_3, x_4)) = [1] + x_4
   POL(if_intlist(x_1, x_2)) = [1] + x_1 + x_2
   POL(int(x_1, x_2)) = [1]
   POL(intlist(x_1)) = [1]
   POL(nil) = [1]
   POL(p(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1
   POL(tail(x_1)) = [1] + x_1
   POL(true) = 0
   POL(zero(x_1)) = 0

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1


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

(155) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) by IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
----------------------------------------

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1


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

(157) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) by 
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, true, 0, 0) -> c16(IF1(true, 0, 0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, true, 0, 0) -> c16(IF1(true, 0, 0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, true, 0, 0) -> c16(IF1(true, 0, 0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1


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

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF_INT(true, true, 0, 0) -> c16
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF_INT(true, true, 0, 0) -> c16
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c16


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

(161) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF_INT(true, true, 0, 0) -> c16
We considered the (Usable) Rules:none
And the Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF_INT(true, true, 0, 0) -> c16
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IF1(x_1, x_2, x_3)) = [1] + x_2
   POL(IF2(x_1, x_2, x_3)) = [1]
   POL(IF_INT(x_1, x_2, x_3, x_4)) = [1] + x_1
   POL(IF_INTLIST(x_1, x_2)) = x_1
   POL(INT(x_1, x_2)) = [1]
   POL(INTLIST(x_1)) = 0
   POL(P(x_1)) = 0
   POL(c10(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16) = 0
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c22(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1
   POL(empty(x_1)) = x_1
   POL(false) = 0
   POL(head(x_1)) = [1] + x_1
   POL(if1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if2(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if_int(x_1, x_2, x_3, x_4)) = [1] + x_4
   POL(if_intlist(x_1, x_2)) = [1] + x_1 + x_2
   POL(int(x_1, x_2)) = [1]
   POL(intlist(x_1)) = [1]
   POL(nil) = [1]
   POL(p(x_1)) = 0
   POL(s(x_1)) = 0
   POL(tail(x_1)) = [1] + x_1
   POL(true) = 0
   POL(zero(x_1)) = 0

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

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF_INT(true, true, 0, 0) -> c16
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF_INT(true, true, 0, 0) -> c16
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c16


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

(163) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
We considered the (Usable) Rules:none
And the Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF_INT(true, true, 0, 0) -> c16
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(IF1(x_1, x_2, x_3)) = [1]
   POL(IF2(x_1, x_2, x_3)) = [1] + x_2
   POL(IF_INT(x_1, x_2, x_3, x_4)) = x_1 + x_3
   POL(IF_INTLIST(x_1, x_2)) = 0
   POL(INT(x_1, x_2)) = [1]
   POL(INTLIST(x_1)) = 0
   POL(P(x_1)) = 0
   POL(c10(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16) = 0
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c22(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1
   POL(empty(x_1)) = x_1
   POL(false) = [1]
   POL(head(x_1)) = [1] + x_1
   POL(if1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if2(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if_int(x_1, x_2, x_3, x_4)) = [1] + x_4
   POL(if_intlist(x_1, x_2)) = [1] + x_1 + x_2
   POL(int(x_1, x_2)) = [1]
   POL(intlist(x_1)) = [1]
   POL(nil) = [1]
   POL(p(x_1)) = 0
   POL(s(x_1)) = 0
   POL(tail(x_1)) = [1] + x_1
   POL(true) = 0
   POL(zero(x_1)) = 0

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF_INT(true, true, 0, 0) -> c16
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF_INT(true, true, 0, 0) -> c16
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c16


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

(165) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) by IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
----------------------------------------

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF_INT(true, true, 0, 0) -> c16
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
K tuples:
   INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0))
   INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF_INT(true, true, 0, 0) -> c16
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c16


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

(167) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IF_INT(true, true, 0, 0) -> c16

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1


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

(169) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) by IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
----------------------------------------

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
S tuples:
   P(s(s(z0))) -> c9(P(s(z0)))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3

Compound Symbols:   c9_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1


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

(171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace P(s(s(z0))) -> c9(P(s(z0))) by 
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
S tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1

Compound Symbols:   c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1


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

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
S tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1

Compound Symbols:   c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1


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(175) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) by IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
----------------------------------------

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
S tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1

Compound Symbols:   c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1


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

(177) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
We considered the (Usable) Rules:
   p(0) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
And the Tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IF1(x_1, x_2, x_3)) = x_2 + x_3
   POL(IF2(x_1, x_2, x_3)) = x_3
   POL(IF_INT(x_1, x_2, x_3, x_4)) = x_1 + x_4
   POL(IF_INTLIST(x_1, x_2)) = x_1
   POL(INT(x_1, x_2)) = x_2
   POL(INTLIST(x_1)) = 0
   POL(P(x_1)) = 0
   POL(c10(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c22(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1
   POL(empty(x_1)) = x_1
   POL(false) = 0
   POL(head(x_1)) = [1] + x_1
   POL(if1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if2(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if_int(x_1, x_2, x_3, x_4)) = [1] + x_4
   POL(if_intlist(x_1, x_2)) = [1] + x_1 + x_2
   POL(int(x_1, x_2)) = [1]
   POL(intlist(x_1)) = [1]
   POL(nil) = [1]
   POL(p(x_1)) = x_1
   POL(s(x_1)) = [1]
   POL(tail(x_1)) = [1] + x_1
   POL(true) = 0
   POL(zero(x_1)) = 0

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

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
S tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1

Compound Symbols:   c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1


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

(179) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) by IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
----------------------------------------

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
S tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0))))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1

Compound Symbols:   c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1


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

(181) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) by IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
----------------------------------------

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
S tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1

Compound Symbols:   c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_2, c21_3, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1


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

(183) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) by IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(p(0)), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
----------------------------------------

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(p(0)), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
S tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1

Compound Symbols:   c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_2, c21_3, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1


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

(185) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) by IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
----------------------------------------

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(p(0)), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
S tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1

Compound Symbols:   c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_2, c21_3, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1


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

(187) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) by IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(0), p(z0))), INT(0, p(z0)))
----------------------------------------

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(p(0)), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(0), p(z0))), INT(0, p(z0)))
S tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1

Compound Symbols:   c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_2, c21_3, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1


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

(189) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) by IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
----------------------------------------

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(p(0)), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(0), p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
S tuples:
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1

Compound Symbols:   c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1


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

(191) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) by 
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, true, s(x0), 0) -> c17(IF2(true, s(x0), 0))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))

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

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(p(0)), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(0), p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, true, s(x0), 0) -> c17(IF2(true, s(x0), 0))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
S tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, true, s(x0), 0) -> c17(IF2(true, s(x0), 0))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1

Compound Symbols:   c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1


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

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

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(p(0)), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(0), p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF_INT(false, true, s(x0), 0) -> c17
S tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF_INT(false, true, s(x0), 0) -> c17
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1))
   IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1

Compound Symbols:   c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c17


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

(195) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF_INT(false, true, s(x0), 0) -> c17
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(P(x1))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
   IF2(false, 0, s(s(x1))) -> c(P(s(s(x1))))
   IF2(false, 0, z0) -> c(INT(0, p(z0)))
S tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF_INT(false, true, s(x0), 0) -> c17
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(P(x1))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
   IF2(false, 0, s(s(x1))) -> c(P(s(s(x1))))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1

Compound Symbols:   c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c17, c_1


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

(197) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   IF2(false, 0, x1) -> c(P(x1))
   IF2(false, 0, s(s(x1))) -> c(P(s(s(x1))))

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

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF_INT(false, true, s(x0), 0) -> c17
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
   IF2(false, 0, z0) -> c(INT(0, p(z0)))
S tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF_INT(false, true, s(x0), 0) -> c17
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1

Compound Symbols:   c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c17, c_1


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

(199) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF_INT(false, true, s(x0), 0) -> c17
We considered the (Usable) Rules:
   zero(0) -> true
   zero(s(z0)) -> false
And the Tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF_INT(false, true, s(x0), 0) -> c17
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
   IF2(false, 0, z0) -> c(INT(0, p(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IF1(x_1, x_2, x_3)) = x_1 + x_2
   POL(IF2(x_1, x_2, x_3)) = x_1
   POL(IF_INT(x_1, x_2, x_3, x_4)) = x_2
   POL(IF_INTLIST(x_1, x_2)) = 0
   POL(INT(x_1, x_2)) = [1]
   POL(INTLIST(x_1)) = 0
   POL(P(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c17) = 0
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c22(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1
   POL(empty(x_1)) = [1] + x_1
   POL(false) = [1]
   POL(head(x_1)) = [1] + x_1
   POL(if1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if2(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if_int(x_1, x_2, x_3, x_4)) = [1] + x_4
   POL(if_intlist(x_1, x_2)) = [1] + x_1 + x_2
   POL(int(x_1, x_2)) = [1]
   POL(intlist(x_1)) = [1]
   POL(nil) = [1]
   POL(p(x_1)) = 0
   POL(s(x_1)) = 0
   POL(tail(x_1)) = [1] + x_1
   POL(true) = [1]
   POL(zero(x_1)) = [1]

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

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF_INT(false, true, s(x0), 0) -> c17
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
   IF2(false, 0, z0) -> c(INT(0, p(z0)))
S tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
   IF_INT(false, true, s(x0), 0) -> c17
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1

Compound Symbols:   c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c17, c_1


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

(201) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) by IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0))
----------------------------------------

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF_INT(false, true, s(x0), 0) -> c17
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
   IF2(false, 0, z0) -> c(INT(0, p(z0)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0))
S tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
   IF_INT(false, true, s(x0), 0) -> c17
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1

Compound Symbols:   c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c17, c_1


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

(203) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IF_INT(false, true, s(x0), 0) -> c17

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

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
   IF2(false, 0, z0) -> c(INT(0, p(z0)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0))
S tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1

Compound Symbols:   c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c_1


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

(205) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) by IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), 0), P(s(s(z0))))
----------------------------------------

(206)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
   IF2(false, 0, z0) -> c(INT(0, p(z0)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), 0), P(s(s(z0))))
S tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, 0) -> c21(INT(p(x0), p(0)))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), 0), P(s(s(z0))))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1

Compound Symbols:   c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c_1


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

(207) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) by IF2(false, z0, 0) -> c21(INT(p(z0), 0))
----------------------------------------

(208)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
   IF2(false, 0, z0) -> c(INT(0, p(z0)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), 0), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INT(p(z0), 0))
S tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), 0), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INT(p(z0), 0))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1

Compound Symbols:   c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c_1


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

(209) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) by IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), 0))
----------------------------------------

(210)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0))))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, 0) -> c14(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, 0) -> c15(IF_INT(true, true, 0, 0))
   INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
   IF2(false, 0, z0) -> c(INT(0, p(z0)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), 0), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INT(p(z0), 0))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), 0))
S tuples:
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1)))
   IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1))
   INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1))
   INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1))
   INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1))
   IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0))
   IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0))))
   IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0))))
   IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0))
   IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0))
   IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0))
   IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0))))
   IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1)))))
   IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))))
   IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0)))))
   IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0))))
   IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)))
   IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0)))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0))))
   IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1))
   IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))))
   IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0)))))
   IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1))))
   IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0))
   IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0)))
   IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)))
   IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))))
   IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1))
   IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1))
   IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1))
   IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1)))
   IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))))
   IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)))
   IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)))
   IF2(false, x0, 0) -> c22(INT(p(x0), p(0)))
   IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))))
   IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0))))
   INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1)))
   INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1)))
   INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0))
   IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0)))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0))
   P(s(s(s(y0)))) -> c9(P(s(s(y0))))
   IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)))
   IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))))
   IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0))
   IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0))))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0)))
   IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0)))
   IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1))
   IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1)))
   IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), 0), P(s(s(z0))))
   IF2(false, z0, 0) -> c21(INT(p(z0), 0))
   IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), 0))
K tuples:
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0))
   IF2(false, 0, x0) -> c21(INT(p(0), p(x0)))
   IF2(false, 0, x0) -> c22(INT(p(0), p(x0)))
   IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0))
   IF2(false, 0, 0) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, 0) -> c(INT(p(0), p(0)))
   IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0)))
   IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0))))
   IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1))))))
   IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1)))))
   IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))))
   IF2(false, 0, x0) -> c(INT(p(0), p(x0)))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))))
   IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))))
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1

Compound Symbols:   c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c_1