KILLED
proof of input_ks3QBOB79e.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 1 ms]
        (14) CpxRNTS
        (15) InliningProof [UPPER BOUND(ID), 70 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), 6 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 346 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 171 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 91 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 192 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 34 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 92 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 14 ms]
        (44) CpxRNTS
        (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 159 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 15 ms]
        (50) CpxRNTS
        (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (52) CpxRNTS
        (53) IntTrsBoundProof [UPPER BOUND(ID), 623 ms]
        (54) CpxRNTS
        (55) IntTrsBoundProof [UPPER BOUND(ID), 222 ms]
        (56) CpxRNTS
        (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (58) CpxRNTS
        (59) IntTrsBoundProof [UPPER BOUND(ID), 100 ms]
        (60) CpxRNTS
        (61) IntTrsBoundProof [UPPER BOUND(ID), 54 ms]
        (62) CpxRNTS
        (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (64) CpxRNTS
        (65) IntTrsBoundProof [UPPER BOUND(ID), 834 ms]
        (66) CpxRNTS
        (67) IntTrsBoundProof [UPPER BOUND(ID), 334 ms]
        (68) CpxRNTS
        (69) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (70) CpxRNTS
        (71) IntTrsBoundProof [UPPER BOUND(ID), 6968 ms]
        (72) CpxRNTS
        (73) IntTrsBoundProof [UPPER BOUND(ID), 1810 ms]
        (74) CpxRNTS
    (75) CompletionProof [UPPER BOUND(ID), 0 ms]
    (76) CpxTypedWeightedCompleteTrs
        (77) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 12 ms]
        (78) CpxRNTS
(79) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(80) CdtProblem
    (81) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 1 ms]
    (86) CdtProblem
    (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 76 ms]
    (124) CdtProblem
    (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 98 ms]
    (130) CdtProblem
    (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 120 ms]
    (134) CdtProblem
    (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 101 ms]
    (138) CdtProblem
    (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 110 ms]
    (166) CdtProblem
    (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem
    (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (176) CdtProblem
    (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (178) CdtProblem
    (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (180) CdtProblem
    (181) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 120 ms]
    (182) CdtProblem
    (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (184) CdtProblem
    (185) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (186) CdtProblem
    (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (188) CdtProblem
    (189) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (190) CdtProblem
    (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (192) CdtProblem
    (193) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (194) CdtProblem
    (195) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 133 ms]
    (196) CdtProblem
    (197) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (198) CdtProblem
    (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (200) CdtProblem
    (201) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (202) CdtProblem
    (203) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (204) CdtProblem
    (205) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (206) CdtProblem
    (207) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (208) CdtProblem
    (209) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (210) CdtProblem
    (211) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (212) CdtProblem
    (213) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (214) CdtProblem
    (215) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (216) CdtProblem
    (217) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (218) CdtProblem
    (219) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (220) CdtProblem
    (221) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (222) CdtProblem
    (223) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (224) CdtProblem
    (225) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (226) CdtProblem
    (227) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (228) CdtProblem
    (229) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 182 ms]
    (230) CdtProblem
    (231) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (232) 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:

   car(cons(x, l)) -> x
   cddr(nil) -> nil
   cddr(cons(x, nil)) -> nil
   cddr(cons(x, cons(y, l))) -> l
   cadr(cons(x, cons(y, l))) -> y
   isZero(0) -> true
   isZero(s(x)) -> false
   plus(x, y) -> ifplus(isZero(x), x, y)
   ifplus(true, x, y) -> y
   ifplus(false, x, y) -> s(plus(p(x), y))
   times(x, y) -> iftimes(isZero(x), x, y)
   iftimes(true, x, y) -> 0
   iftimes(false, x, y) -> plus(y, times(p(x), y))
   p(s(x)) -> x
   p(0) -> 0
   shorter(nil, y) -> true
   shorter(cons(x, l), 0) -> false
   shorter(cons(x, l), s(y)) -> shorter(l, y)
   prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l)
   if(true, b, l) -> s(0)
   if(false, b, l) -> if2(b, l)
   if2(true, l) -> car(l)
   if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

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:

   car(cons(x, l)) -> x
   cddr(nil) -> nil
   cddr(cons(x, nil)) -> nil
   cddr(cons(x, cons(y, l))) -> l
   cadr(cons(x, cons(y, l))) -> y
   isZero(0') -> true
   isZero(s(x)) -> false
   plus(x, y) -> ifplus(isZero(x), x, y)
   ifplus(true, x, y) -> y
   ifplus(false, x, y) -> s(plus(p(x), y))
   times(x, y) -> iftimes(isZero(x), x, y)
   iftimes(true, x, y) -> 0'
   iftimes(false, x, y) -> plus(y, times(p(x), y))
   p(s(x)) -> x
   p(0') -> 0'
   shorter(nil, y) -> true
   shorter(cons(x, l), 0') -> false
   shorter(cons(x, l), s(y)) -> shorter(l, y)
   prod(l) -> if(shorter(l, 0'), shorter(l, s(0')), l)
   if(true, b, l) -> s(0')
   if(false, b, l) -> if2(b, l)
   if2(true, l) -> car(l)
   if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

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:

   car(cons(x, l)) -> x
   cddr(nil) -> nil
   cddr(cons(x, nil)) -> nil
   cddr(cons(x, cons(y, l))) -> l
   cadr(cons(x, cons(y, l))) -> y
   isZero(0) -> true
   isZero(s(x)) -> false
   plus(x, y) -> ifplus(isZero(x), x, y)
   ifplus(true, x, y) -> y
   ifplus(false, x, y) -> s(plus(p(x), y))
   times(x, y) -> iftimes(isZero(x), x, y)
   iftimes(true, x, y) -> 0
   iftimes(false, x, y) -> plus(y, times(p(x), y))
   p(s(x)) -> x
   p(0) -> 0
   shorter(nil, y) -> true
   shorter(cons(x, l), 0) -> false
   shorter(cons(x, l), s(y)) -> shorter(l, y)
   prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l)
   if(true, b, l) -> s(0)
   if(false, b, l) -> if2(b, l)
   if2(true, l) -> car(l)
   if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

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:

   car(cons(x, l)) -> x [1]
   cddr(nil) -> nil [1]
   cddr(cons(x, nil)) -> nil [1]
   cddr(cons(x, cons(y, l))) -> l [1]
   cadr(cons(x, cons(y, l))) -> y [1]
   isZero(0) -> true [1]
   isZero(s(x)) -> false [1]
   plus(x, y) -> ifplus(isZero(x), x, y) [1]
   ifplus(true, x, y) -> y [1]
   ifplus(false, x, y) -> s(plus(p(x), y)) [1]
   times(x, y) -> iftimes(isZero(x), x, y) [1]
   iftimes(true, x, y) -> 0 [1]
   iftimes(false, x, y) -> plus(y, times(p(x), y)) [1]
   p(s(x)) -> x [1]
   p(0) -> 0 [1]
   shorter(nil, y) -> true [1]
   shorter(cons(x, l), 0) -> false [1]
   shorter(cons(x, l), s(y)) -> shorter(l, y) [1]
   prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l) [1]
   if(true, b, l) -> s(0) [1]
   if(false, b, l) -> if2(b, l) [1]
   if2(true, l) -> car(l) [1]
   if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l))) [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:

   car(cons(x, l)) -> x [1]
   cddr(nil) -> nil [1]
   cddr(cons(x, nil)) -> nil [1]
   cddr(cons(x, cons(y, l))) -> l [1]
   cadr(cons(x, cons(y, l))) -> y [1]
   isZero(0) -> true [1]
   isZero(s(x)) -> false [1]
   plus(x, y) -> ifplus(isZero(x), x, y) [1]
   ifplus(true, x, y) -> y [1]
   ifplus(false, x, y) -> s(plus(p(x), y)) [1]
   times(x, y) -> iftimes(isZero(x), x, y) [1]
   iftimes(true, x, y) -> 0 [1]
   iftimes(false, x, y) -> plus(y, times(p(x), y)) [1]
   p(s(x)) -> x [1]
   p(0) -> 0 [1]
   shorter(nil, y) -> true [1]
   shorter(cons(x, l), 0) -> false [1]
   shorter(cons(x, l), s(y)) -> shorter(l, y) [1]
   prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l) [1]
   if(true, b, l) -> s(0) [1]
   if(false, b, l) -> if2(b, l) [1]
   if2(true, l) -> car(l) [1]
   if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l))) [1]

The TRS has the following type information:
car :: cons:nil -> 0:s
cons :: 0:s -> cons:nil -> cons:nil
cddr :: cons:nil -> cons:nil
nil :: cons:nil
cadr :: cons:nil -> 0:s
isZero :: 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
plus :: 0:s -> 0:s -> 0:s
ifplus :: true:false -> 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
iftimes :: true:false -> 0:s -> 0:s -> 0:s
shorter :: cons:nil -> 0:s -> true:false
prod :: cons:nil -> 0:s
if :: true:false -> true:false -> cons:nil -> 0:s
if2 :: true:false -> cons:nil -> 0:s

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:

   prod_1
   if_3
   if2_2

(c) The following functions are completely defined:

   times_2
   car_1
   cadr_1
   cddr_1
   p_1
   isZero_1
   shorter_2
   iftimes_3
   plus_2
   ifplus_3

Due to the following rules being added:

   car(v0) -> 0 [0]
   cadr(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:

   car(cons(x, l)) -> x [1]
   cddr(nil) -> nil [1]
   cddr(cons(x, nil)) -> nil [1]
   cddr(cons(x, cons(y, l))) -> l [1]
   cadr(cons(x, cons(y, l))) -> y [1]
   isZero(0) -> true [1]
   isZero(s(x)) -> false [1]
   plus(x, y) -> ifplus(isZero(x), x, y) [1]
   ifplus(true, x, y) -> y [1]
   ifplus(false, x, y) -> s(plus(p(x), y)) [1]
   times(x, y) -> iftimes(isZero(x), x, y) [1]
   iftimes(true, x, y) -> 0 [1]
   iftimes(false, x, y) -> plus(y, times(p(x), y)) [1]
   p(s(x)) -> x [1]
   p(0) -> 0 [1]
   shorter(nil, y) -> true [1]
   shorter(cons(x, l), 0) -> false [1]
   shorter(cons(x, l), s(y)) -> shorter(l, y) [1]
   prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l) [1]
   if(true, b, l) -> s(0) [1]
   if(false, b, l) -> if2(b, l) [1]
   if2(true, l) -> car(l) [1]
   if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l))) [1]
   car(v0) -> 0 [0]
   cadr(v0) -> 0 [0]

The TRS has the following type information:
car :: cons:nil -> 0:s
cons :: 0:s -> cons:nil -> cons:nil
cddr :: cons:nil -> cons:nil
nil :: cons:nil
cadr :: cons:nil -> 0:s
isZero :: 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
plus :: 0:s -> 0:s -> 0:s
ifplus :: true:false -> 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
iftimes :: true:false -> 0:s -> 0:s -> 0:s
shorter :: cons:nil -> 0:s -> true:false
prod :: cons:nil -> 0:s
if :: true:false -> true:false -> cons:nil -> 0:s
if2 :: true:false -> cons:nil -> 0:s

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:

   car(cons(x, l)) -> x [1]
   cddr(nil) -> nil [1]
   cddr(cons(x, nil)) -> nil [1]
   cddr(cons(x, cons(y, l))) -> l [1]
   cadr(cons(x, cons(y, l))) -> y [1]
   isZero(0) -> true [1]
   isZero(s(x)) -> false [1]
   plus(0, y) -> ifplus(true, 0, y) [2]
   plus(s(x'), y) -> ifplus(false, s(x'), y) [2]
   ifplus(true, x, y) -> y [1]
   ifplus(false, s(x''), y) -> s(plus(x'', y)) [2]
   ifplus(false, 0, y) -> s(plus(0, y)) [2]
   times(0, y) -> iftimes(true, 0, y) [2]
   times(s(x1), y) -> iftimes(false, s(x1), y) [2]
   iftimes(true, x, y) -> 0 [1]
   iftimes(false, s(x2), y) -> plus(y, times(x2, y)) [2]
   iftimes(false, 0, y) -> plus(y, times(0, y)) [2]
   p(s(x)) -> x [1]
   p(0) -> 0 [1]
   shorter(nil, y) -> true [1]
   shorter(cons(x, l), 0) -> false [1]
   shorter(cons(x, l), s(y)) -> shorter(l, y) [1]
   prod(nil) -> if(true, true, nil) [3]
   prod(cons(x3, l')) -> if(false, shorter(l', 0), cons(x3, l')) [3]
   if(true, b, l) -> s(0) [1]
   if(false, b, l) -> if2(b, l) [1]
   if2(true, l) -> car(l) [1]
   if2(false, cons(x4, cons(y', l1))) -> prod(cons(times(x4, y'), l1)) [4]
   if2(false, cons(x4, nil)) -> prod(cons(times(x4, 0), nil)) [3]
   if2(false, cons(x4, cons(y1, l3))) -> prod(cons(times(x4, 0), l3)) [3]
   if2(false, cons(x5, cons(y'', l2))) -> prod(cons(times(0, y''), l2)) [3]
   if2(false, nil) -> prod(cons(times(0, 0), nil)) [2]
   if2(false, cons(x6, nil)) -> prod(cons(times(0, 0), nil)) [2]
   if2(false, cons(x7, cons(y2, l4))) -> prod(cons(times(0, 0), l4)) [2]
   car(v0) -> 0 [0]
   cadr(v0) -> 0 [0]

The TRS has the following type information:
car :: cons:nil -> 0:s
cons :: 0:s -> cons:nil -> cons:nil
cddr :: cons:nil -> cons:nil
nil :: cons:nil
cadr :: cons:nil -> 0:s
isZero :: 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
plus :: 0:s -> 0:s -> 0:s
ifplus :: true:false -> 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
iftimes :: true:false -> 0:s -> 0:s -> 0:s
shorter :: cons:nil -> 0:s -> true:false
prod :: cons:nil -> 0:s
if :: true:false -> true:false -> cons:nil -> 0:s
if2 :: true:false -> cons:nil -> 0:s

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

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

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

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x + 0
   if(z, z', z'') -{ 1 }-> if2(b, l) :|: b >= 0, z' = b, l >= 0, z = 0, z'' = l
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: b >= 0, z = 1, z' = b, l >= 0, z'' = l
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + 0) :|: x4 >= 0, z = 0, z' = 1 + x4 + 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' = 1 + x6 + 0, x6 >= 0, z = 0
   if2(z, z') -{ 1 }-> car(l) :|: z' = l, z = 1, l >= 0
   ifplus(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(x'', y) :|: z' = 1 + x'', z'' = y, y >= 0, x'' >= 0, z = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, y) :|: z'' = y, y >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(y, times(x2, y)) :|: z' = 1 + x2, z'' = y, y >= 0, z = 0, x2 >= 0
   iftimes(z, z', z'') -{ 2 }-> plus(y, times(0, y)) :|: z'' = y, y >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, y) :|: y >= 0, z = 0, z' = y
   plus(z, z') -{ 2 }-> ifplus(0, 1 + x', y) :|: z = 1 + x', x' >= 0, y >= 0, z' = y
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 3 }-> if(0, shorter(l', 0), 1 + x3 + l') :|: l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 }-> shorter(l, y) :|: z' = 1 + y, x >= 0, y >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, y) :|: y >= 0, z = 0, z' = y
   times(z, z') -{ 2 }-> iftimes(0, 1 + x1, y) :|: x1 >= 0, y >= 0, z = 1 + x1, z' = y


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

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

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

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

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

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x + 0
   if(z, z', z'') -{ 1 }-> if2(b, l) :|: b >= 0, z' = b, l >= 0, z = 0, z'' = l
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: b >= 0, z = 1, z' = b, l >= 0, z'' = l
   if2(z, z') -{ 2 }-> x :|: z' = l, z = 1, l >= 0, x >= 0, l' >= 0, l = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + 0) :|: x4 >= 0, z = 0, z' = 1 + x4 + 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' = 1 + x6 + 0, x6 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z' = l, z = 1, l >= 0, v0 >= 0, l = v0
   ifplus(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(x'', y) :|: z' = 1 + x'', z'' = y, y >= 0, x'' >= 0, z = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, y) :|: z'' = y, y >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(y, times(x2, y)) :|: z' = 1 + x2, z'' = y, y >= 0, z = 0, x2 >= 0
   iftimes(z, z', z'') -{ 2 }-> plus(y, times(0, y)) :|: z'' = y, y >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, y) :|: y >= 0, z = 0, z' = y
   plus(z, z') -{ 2 }-> ifplus(0, 1 + x', y) :|: z = 1 + x', x' >= 0, y >= 0, z' = y
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 3 }-> if(0, shorter(l', 0), 1 + x3 + l') :|: l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 }-> shorter(l, y) :|: z' = 1 + y, x >= 0, y >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, y) :|: y >= 0, z = 0, z' = y
   times(z, z') -{ 2 }-> iftimes(0, 1 + x1, y) :|: x1 >= 0, y >= 0, z = 1 + x1, z' = y


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

(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:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 3 }-> if(0, shorter(l', 0), 1 + x3 + l') :|: l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 }-> shorter(l, z' - 1) :|: x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0


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

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

   { shorter }
   { isZero }
   { cddr }
   { car }
   { cadr }
   { plus, ifplus }
   { p }
   { iftimes, times }
   { if2, if, prod }

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

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

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 3 }-> if(0, shorter(l', 0), 1 + x3 + l') :|: l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 }-> shorter(l, z' - 1) :|: x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {shorter}, {isZero}, {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}

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

(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:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 3 }-> if(0, shorter(l', 0), 1 + x3 + l') :|: l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 }-> shorter(l, z' - 1) :|: x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {shorter}, {isZero}, {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: shorter
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:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 3 }-> if(0, shorter(l', 0), 1 + x3 + l') :|: l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 }-> shorter(l, z' - 1) :|: x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {shorter}, {isZero}, {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: ?, size: O(1) [1]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 3 }-> if(0, shorter(l', 0), 1 + x3 + l') :|: l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 }-> shorter(l, z' - 1) :|: x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {isZero}, {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], 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:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {isZero}, {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: isZero
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:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {isZero}, {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: ?, size: O(1) [1]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: isZero
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:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: 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:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: cddr
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:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {cddr}, {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: ?, size: O(n^1) [z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: cddr
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:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: 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:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: car
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:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {car}, {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: ?, size: O(n^1) [z]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: car
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:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: 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:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: O(1) [1], size: O(n^1) [z]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {cadr}, {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: O(1) [1], size: O(n^1) [z]
  cadr: runtime: ?, size: O(n^1) [z]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: O(1) [1], size: O(n^1) [z]
  cadr: runtime: O(1) [1], size: O(n^1) [z]

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

(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:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: O(1) [1], size: O(n^1) [z]
  cadr: runtime: O(1) [1], size: O(n^1) [z]

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus,ifplus}, {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: O(1) [1], size: O(n^1) [z]
  cadr: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: ?, size: O(n^1) [z + z']
  ifplus: runtime: ?, size: O(n^1) [1 + z' + z'']

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 6 + 4*z

Computed RUNTIME bound using CoFloCo for: ifplus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 8 + 4*z'

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

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

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   ifplus(z, z', z'') -{ 2 }-> 1 + plus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 2 }-> ifplus(1, 0, z') :|: z' >= 0, z = 0
   plus(z, z') -{ 2 }-> ifplus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: O(1) [1], size: O(n^1) [z]
  cadr: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z']
  ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + 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:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0
   ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0
   plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: O(1) [1], size: O(n^1) [z]
  cadr: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z']
  ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z'']

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

(59) 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

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

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

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0
   ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0
   plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {p}, {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: O(1) [1], size: O(n^1) [z]
  cadr: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z']
  ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z'']
  p: runtime: ?, size: O(n^1) [z]

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0
   ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0
   plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: O(1) [1], size: O(n^1) [z]
  cadr: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z']
  ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z]

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

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

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0
   ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0
   plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: O(1) [1], size: O(n^1) [z]
  cadr: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z']
  ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z]

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

(65) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0
   ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0
   plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {iftimes,times}, {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: O(1) [1], size: O(n^1) [z]
  cadr: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z']
  ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z]
  iftimes: runtime: ?, size: O(n^2) [z'*z'' + 2*z'']
  times: runtime: ?, size: O(n^2) [z*z' + 4*z']

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

(67) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: iftimes
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 21 + 10*z' + 4*z'*z'' + 8*z''

Computed RUNTIME bound using KoAT for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 46 + 10*z + 4*z*z' + 16*z'

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

(68)
Obligation:
Complexity RNTS consisting of the following rules:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 4 }-> prod(1 + times(x4, y') + l1) :|: z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(x4, 0) + l3) :|: y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 3 }-> prod(1 + times(0, y'') + l2) :|: x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + l4) :|: x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z = 0, z' = 0
   if2(z, z') -{ 2 }-> prod(1 + times(0, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 3 }-> prod(1 + times(z' - 1, 0) + 0) :|: z' - 1 >= 0, z = 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0
   ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 2 }-> plus(z'', times(z' - 1, z'')) :|: z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0
   plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 2 }-> iftimes(1, 0, z') :|: z' >= 0, z = 0
   times(z, z') -{ 2 }-> iftimes(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: O(1) [1], size: O(n^1) [z]
  cadr: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z']
  ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z]
  iftimes: runtime: O(n^2) [21 + 10*z' + 4*z'*z'' + 8*z''], size: O(n^2) [z'*z'' + 2*z'']
  times: runtime: O(n^2) [46 + 10*z + 4*z*z' + 16*z'], size: O(n^2) [z*z' + 4*z']

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

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

(70)
Obligation:
Complexity RNTS consisting of the following rules:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 50 + 10*x4 + 4*x4*y' + 16*y' }-> prod(1 + s10 + l1) :|: s10 >= 0, s10 <= y' * x4 + 4 * y', z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 39 + 10*z' }-> prod(1 + s11 + 0) :|: s11 >= 0, s11 <= 0 * (z' - 1) + 4 * 0, z' - 1 >= 0, z = 0
   if2(z, z') -{ 49 + 10*x4 }-> prod(1 + s12 + l3) :|: s12 >= 0, s12 <= 0 * x4 + 4 * 0, y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 49 + 16*y'' }-> prod(1 + s13 + l2) :|: s13 >= 0, s13 <= y'' * 0 + 4 * y'', x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 48 }-> prod(1 + s14 + 0) :|: s14 >= 0, s14 <= 0 * 0 + 4 * 0, z = 0, z' = 0
   if2(z, z') -{ 48 }-> prod(1 + s15 + 0) :|: s15 >= 0, s15 <= 0 * 0 + 4 * 0, z' - 1 >= 0, z = 0
   if2(z, z') -{ 48 }-> prod(1 + s16 + l4) :|: s16 >= 0, s16 <= 0 * 0 + 4 * 0, x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0
   ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 44 + 10*z' + 4*z'*z'' + 16*z'' }-> s7 :|: s6 >= 0, s6 <= z'' * (z' - 1) + 4 * z'', s7 >= 0, s7 <= z'' + s6, z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 54 + 20*z'' }-> s9 :|: s8 >= 0, s8 <= z'' * 0 + 4 * z'', s9 >= 0, s9 <= z'' + s8, z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0
   plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 23 + 8*z' }-> s4 :|: s4 >= 0, s4 <= 0 * z' + 2 * z', z' >= 0, z = 0
   times(z, z') -{ 23 + 10*z + 4*z*z' + 8*z' }-> s5 :|: s5 >= 0, s5 <= (1 + (z - 1)) * z' + 2 * z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: O(1) [1], size: O(n^1) [z]
  cadr: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z']
  ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z]
  iftimes: runtime: O(n^2) [21 + 10*z' + 4*z'*z'' + 8*z''], size: O(n^2) [z'*z'' + 2*z'']
  times: runtime: O(n^2) [46 + 10*z + 4*z*z' + 16*z'], size: O(n^2) [z*z' + 4*z']

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

(71) 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: if
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 50 + 10*x4 + 4*x4*y' + 16*y' }-> prod(1 + s10 + l1) :|: s10 >= 0, s10 <= y' * x4 + 4 * y', z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 39 + 10*z' }-> prod(1 + s11 + 0) :|: s11 >= 0, s11 <= 0 * (z' - 1) + 4 * 0, z' - 1 >= 0, z = 0
   if2(z, z') -{ 49 + 10*x4 }-> prod(1 + s12 + l3) :|: s12 >= 0, s12 <= 0 * x4 + 4 * 0, y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 49 + 16*y'' }-> prod(1 + s13 + l2) :|: s13 >= 0, s13 <= y'' * 0 + 4 * y'', x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 48 }-> prod(1 + s14 + 0) :|: s14 >= 0, s14 <= 0 * 0 + 4 * 0, z = 0, z' = 0
   if2(z, z') -{ 48 }-> prod(1 + s15 + 0) :|: s15 >= 0, s15 <= 0 * 0 + 4 * 0, z' - 1 >= 0, z = 0
   if2(z, z') -{ 48 }-> prod(1 + s16 + l4) :|: s16 >= 0, s16 <= 0 * 0 + 4 * 0, x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0
   ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 44 + 10*z' + 4*z'*z'' + 16*z'' }-> s7 :|: s6 >= 0, s6 <= z'' * (z' - 1) + 4 * z'', s7 >= 0, s7 <= z'' + s6, z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 54 + 20*z'' }-> s9 :|: s8 >= 0, s8 <= z'' * 0 + 4 * z'', s9 >= 0, s9 <= z'' + s8, z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0
   plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 23 + 8*z' }-> s4 :|: s4 >= 0, s4 <= 0 * z' + 2 * z', z' >= 0, z = 0
   times(z, z') -{ 23 + 10*z + 4*z*z' + 8*z' }-> s5 :|: s5 >= 0, s5 <= (1 + (z - 1)) * z' + 2 * z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: O(1) [1], size: O(n^1) [z]
  cadr: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z']
  ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z]
  iftimes: runtime: O(n^2) [21 + 10*z' + 4*z'*z'' + 8*z''], size: O(n^2) [z'*z'' + 2*z'']
  times: runtime: O(n^2) [46 + 10*z + 4*z*z' + 16*z'], size: O(n^2) [z*z' + 4*z']
  if2: runtime: ?, size: INF
  if: runtime: ?, size: INF
  prod: runtime: ?, size: INF

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(74)
Obligation:
Complexity RNTS consisting of the following rules:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: z >= 0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: z >= 0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: z - 1 >= 0
   if(z, z', z'') -{ 1 }-> if2(z', z'') :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 1, z'' >= 0
   if2(z, z') -{ 2 }-> x :|: z = 1, z' >= 0, x >= 0, l' >= 0, z' = 1 + x + l'
   if2(z, z') -{ 50 + 10*x4 + 4*x4*y' + 16*y' }-> prod(1 + s10 + l1) :|: s10 >= 0, s10 <= y' * x4 + 4 * y', z' = 1 + x4 + (1 + y' + l1), x4 >= 0, y' >= 0, l1 >= 0, z = 0
   if2(z, z') -{ 39 + 10*z' }-> prod(1 + s11 + 0) :|: s11 >= 0, s11 <= 0 * (z' - 1) + 4 * 0, z' - 1 >= 0, z = 0
   if2(z, z') -{ 49 + 10*x4 }-> prod(1 + s12 + l3) :|: s12 >= 0, s12 <= 0 * x4 + 4 * 0, y1 >= 0, x4 >= 0, z' = 1 + x4 + (1 + y1 + l3), z = 0, l3 >= 0
   if2(z, z') -{ 49 + 16*y'' }-> prod(1 + s13 + l2) :|: s13 >= 0, s13 <= y'' * 0 + 4 * y'', x5 >= 0, z' = 1 + x5 + (1 + y'' + l2), y'' >= 0, z = 0, l2 >= 0
   if2(z, z') -{ 48 }-> prod(1 + s14 + 0) :|: s14 >= 0, s14 <= 0 * 0 + 4 * 0, z = 0, z' = 0
   if2(z, z') -{ 48 }-> prod(1 + s15 + 0) :|: s15 >= 0, s15 <= 0 * 0 + 4 * 0, z' - 1 >= 0, z = 0
   if2(z, z') -{ 48 }-> prod(1 + s16 + l4) :|: s16 >= 0, s16 <= 0 * 0 + 4 * 0, x7 >= 0, z' = 1 + x7 + (1 + y2 + l4), y2 >= 0, z = 0, l4 >= 0
   if2(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0
   ifplus(z, z', z'') -{ 1 }-> z'' :|: z = 1, z' >= 0, z'' >= 0
   ifplus(z, z', z'') -{ 4 + 4*z' }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + z'', z'' >= 0, z' - 1 >= 0, z = 0
   ifplus(z, z', z'') -{ 8 }-> 1 + s3 :|: s3 >= 0, s3 <= 0 + z'', z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 44 + 10*z' + 4*z'*z'' + 16*z'' }-> s7 :|: s6 >= 0, s6 <= z'' * (z' - 1) + 4 * z'', s7 >= 0, s7 <= z'' + s6, z'' >= 0, z = 0, z' - 1 >= 0
   iftimes(z, z', z'') -{ 54 + 20*z'' }-> s9 :|: s8 >= 0, s8 <= z'' * 0 + 4 * z'', s9 >= 0, s9 <= z'' + s8, z'' >= 0, z = 0, z' = 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z - 1 >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 10 }-> s'' :|: s'' >= 0, s'' <= z' + 1 + 0, z' >= 0, z = 0
   plus(z, z') -{ 10 + 4*z }-> s1 :|: s1 >= 0, s1 <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   prod(z) -{ 3 }-> if(1, 1, 0) :|: z = 0
   prod(z) -{ 4 }-> if(0, s', 1 + x3 + l') :|: s' >= 0, s' <= 1, l' >= 0, z = 1 + x3 + l', x3 >= 0
   shorter(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 1, x >= 0, z' - 1 >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   shorter(z, z') -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   times(z, z') -{ 23 + 8*z' }-> s4 :|: s4 >= 0, s4 <= 0 * z' + 2 * z', z' >= 0, z = 0
   times(z, z') -{ 23 + 10*z + 4*z*z' + 8*z' }-> s5 :|: s5 >= 0, s5 <= (1 + (z - 1)) * z' + 2 * z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {if2,if,prod}
Previous analysis results are:
  shorter: runtime: O(n^1) [1 + z'], size: O(1) [1]
  isZero: runtime: O(1) [1], size: O(1) [1]
  cddr: runtime: O(1) [1], size: O(n^1) [z]
  car: runtime: O(1) [1], size: O(n^1) [z]
  cadr: runtime: O(1) [1], size: O(n^1) [z]
  plus: runtime: O(n^1) [6 + 4*z], size: O(n^1) [z + z']
  ifplus: runtime: O(n^1) [8 + 4*z'], size: O(n^1) [1 + z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z]
  iftimes: runtime: O(n^2) [21 + 10*z' + 4*z'*z'' + 8*z''], size: O(n^2) [z'*z'' + 2*z'']
  times: runtime: O(n^2) [46 + 10*z + 4*z*z' + 16*z'], size: O(n^2) [z*z' + 4*z']
  if2: runtime: INF, size: INF
  if: runtime: ?, size: INF
  prod: runtime: ?, size: INF

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

(75) 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:

   car(v0) -> null_car [0]
   cadr(v0) -> null_cadr [0]
   isZero(v0) -> null_isZero [0]
   p(v0) -> null_p [0]
   shorter(v0, v1) -> null_shorter [0]
   ifplus(v0, v1, v2) -> null_ifplus [0]
   iftimes(v0, v1, v2) -> null_iftimes [0]
   if(v0, v1, v2) -> null_if [0]
   if2(v0, v1) -> null_if2 [0]

And the following fresh constants:    null_car, null_cadr, null_isZero, null_p, null_shorter, null_ifplus, null_iftimes, null_if, null_if2

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

(76)
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:

   car(cons(x, l)) -> x [1]
   cddr(nil) -> nil [1]
   cddr(cons(x, nil)) -> nil [1]
   cddr(cons(x, cons(y, l))) -> l [1]
   cadr(cons(x, cons(y, l))) -> y [1]
   isZero(0) -> true [1]
   isZero(s(x)) -> false [1]
   plus(x, y) -> ifplus(isZero(x), x, y) [1]
   ifplus(true, x, y) -> y [1]
   ifplus(false, x, y) -> s(plus(p(x), y)) [1]
   times(x, y) -> iftimes(isZero(x), x, y) [1]
   iftimes(true, x, y) -> 0 [1]
   iftimes(false, x, y) -> plus(y, times(p(x), y)) [1]
   p(s(x)) -> x [1]
   p(0) -> 0 [1]
   shorter(nil, y) -> true [1]
   shorter(cons(x, l), 0) -> false [1]
   shorter(cons(x, l), s(y)) -> shorter(l, y) [1]
   prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l) [1]
   if(true, b, l) -> s(0) [1]
   if(false, b, l) -> if2(b, l) [1]
   if2(true, l) -> car(l) [1]
   if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l))) [1]
   car(v0) -> null_car [0]
   cadr(v0) -> null_cadr [0]
   isZero(v0) -> null_isZero [0]
   p(v0) -> null_p [0]
   shorter(v0, v1) -> null_shorter [0]
   ifplus(v0, v1, v2) -> null_ifplus [0]
   iftimes(v0, v1, v2) -> null_iftimes [0]
   if(v0, v1, v2) -> null_if [0]
   if2(v0, v1) -> null_if2 [0]

The TRS has the following type information:
car :: cons:nil -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
cons :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> cons:nil -> cons:nil
cddr :: cons:nil -> cons:nil
nil :: cons:nil
cadr :: cons:nil -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
isZero :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> true:false:null_isZero:null_shorter
0 :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
true :: true:false:null_isZero:null_shorter
s :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
false :: true:false:null_isZero:null_shorter
plus :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
ifplus :: true:false:null_isZero:null_shorter -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
p :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
times :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
iftimes :: true:false:null_isZero:null_shorter -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
shorter :: cons:nil -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2 -> true:false:null_isZero:null_shorter
prod :: cons:nil -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
if :: true:false:null_isZero:null_shorter -> true:false:null_isZero:null_shorter -> cons:nil -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
if2 :: true:false:null_isZero:null_shorter -> cons:nil -> 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
null_car :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
null_cadr :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
null_isZero :: true:false:null_isZero:null_shorter
null_p :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
null_shorter :: true:false:null_isZero:null_shorter
null_ifplus :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
null_iftimes :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
null_if :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2
null_if2 :: 0:s:null_car:null_cadr:null_p:null_ifplus:null_iftimes:null_if:null_if2

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

(77) 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
   0 => 0
   true => 2
   false => 1
   null_car => 0
   null_cadr => 0
   null_isZero => 0
   null_p => 0
   null_shorter => 0
   null_ifplus => 0
   null_iftimes => 0
   null_if => 0
   null_if2 => 0

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

(78)
Obligation:
Complexity RNTS consisting of the following rules:

   cadr(z) -{ 1 }-> y :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cadr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   car(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   car(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   cddr(z) -{ 1 }-> l :|: x >= 0, y >= 0, l >= 0, z = 1 + x + (1 + y + l)
   cddr(z) -{ 1 }-> 0 :|: z = 0
   cddr(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x + 0
   if(z, z', z'') -{ 1 }-> if2(b, l) :|: b >= 0, z = 1, z' = b, l >= 0, z'' = l
   if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 2, b >= 0, z' = b, l >= 0, z'' = l
   if2(z, z') -{ 1 }-> prod(1 + times(car(l), cadr(l)) + cddr(l)) :|: z' = l, z = 1, l >= 0
   if2(z, z') -{ 1 }-> car(l) :|: z = 2, z' = l, l >= 0
   if2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ifplus(z, z', z'') -{ 1 }-> y :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0
   ifplus(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   ifplus(z, z', z'') -{ 1 }-> 1 + plus(p(x), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   iftimes(z, z', z'') -{ 1 }-> plus(y, times(p(x), y)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   iftimes(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0
   iftimes(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   isZero(z) -{ 1 }-> 2 :|: z = 0
   isZero(z) -{ 1 }-> 1 :|: x >= 0, z = 1 + x
   isZero(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   plus(z, z') -{ 1 }-> ifplus(isZero(x), x, y) :|: x >= 0, y >= 0, z = x, z' = y
   prod(z) -{ 1 }-> if(shorter(l, 0), shorter(l, 1 + 0), l) :|: z = l, l >= 0
   shorter(z, z') -{ 1 }-> shorter(l, y) :|: z' = 1 + y, x >= 0, y >= 0, l >= 0, z = 1 + x + l
   shorter(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y
   shorter(z, z') -{ 1 }-> 1 :|: x >= 0, l >= 0, z = 1 + x + l, z' = 0
   shorter(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   times(z, z') -{ 1 }-> iftimes(isZero(x), x, y) :|: x >= 0, y >= 0, z = x, z' = y

Only complete derivations are relevant for the runtime complexity.

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

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   car(cons(z0, z1)) -> z0
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   cadr(cons(z0, cons(z1, z2))) -> z1
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   p(s(z0)) -> z0
   p(0) -> 0
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   prod(z0) -> if(shorter(z0, 0), shorter(z0, s(0)), z0)
   if(true, z0, z1) -> s(0)
   if(false, z0, z1) -> if2(z0, z1)
   if2(true, z0) -> car(z0)
   if2(false, z0) -> prod(cons(times(car(z0), cadr(z0)), cddr(z0)))
Tuples:
   CAR(cons(z0, z1)) -> c
   CDDR(nil) -> c1
   CDDR(cons(z0, nil)) -> c2
   CDDR(cons(z0, cons(z1, z2))) -> c3
   CADR(cons(z0, cons(z1, z2))) -> c4
   ISZERO(0) -> c5
   ISZERO(s(z0)) -> c6
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1), ISZERO(z0))
   IFPLUS(true, z0, z1) -> c8
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1), P(z0))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1), ISZERO(z0))
   IFTIMES(true, z0, z1) -> c11
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1), P(z0))
   P(s(z0)) -> c13
   P(0) -> c14
   SHORTER(nil, z0) -> c15
   SHORTER(cons(z0, z1), 0) -> c16
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, 0))
   PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0)))
   IF(true, z0, z1) -> c20
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(true, z0) -> c22(CAR(z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CAR(z0))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CADR(z0))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), CDDR(z0))
S tuples:
   CAR(cons(z0, z1)) -> c
   CDDR(nil) -> c1
   CDDR(cons(z0, nil)) -> c2
   CDDR(cons(z0, cons(z1, z2))) -> c3
   CADR(cons(z0, cons(z1, z2))) -> c4
   ISZERO(0) -> c5
   ISZERO(s(z0)) -> c6
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1), ISZERO(z0))
   IFPLUS(true, z0, z1) -> c8
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1), P(z0))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1), ISZERO(z0))
   IFTIMES(true, z0, z1) -> c11
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1), P(z0))
   P(s(z0)) -> c13
   P(0) -> c14
   SHORTER(nil, z0) -> c15
   SHORTER(cons(z0, z1), 0) -> c16
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, 0))
   PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0)))
   IF(true, z0, z1) -> c20
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(true, z0) -> c22(CAR(z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CAR(z0))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CADR(z0))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), CDDR(z0))
K tuples:none
Defined Rule Symbols:   car_1, cddr_1, cadr_1, isZero_1, plus_2, ifplus_3, times_2, iftimes_3, p_1, shorter_2, prod_1, if_3, if2_2

Defined Pair Symbols:   CAR_1, CDDR_1, CADR_1, ISZERO_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, P_1, SHORTER_2, PROD_1, IF_3, IF2_2

Compound Symbols:   c, c1, c2, c3, c4, c5, c6, c7_2, c8, c9_2, c10_2, c11, c12_3, c13, c14, c15, c16, c17_1, c18_2, c19_2, c20, c21_1, c22_1, c23_3, c24_3, c25_2


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

(81) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 15 trailing nodes:
   IF(true, z0, z1) -> c20
   IFTIMES(true, z0, z1) -> c11
   CADR(cons(z0, cons(z1, z2))) -> c4
   SHORTER(nil, z0) -> c15
   ISZERO(0) -> c5
   CDDR(cons(z0, cons(z1, z2))) -> c3
   IF2(true, z0) -> c22(CAR(z0))
   P(s(z0)) -> c13
   CDDR(nil) -> c1
   CDDR(cons(z0, nil)) -> c2
   SHORTER(cons(z0, z1), 0) -> c16
   IFPLUS(true, z0, z1) -> c8
   ISZERO(s(z0)) -> c6
   CAR(cons(z0, z1)) -> c
   P(0) -> c14

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   car(cons(z0, z1)) -> z0
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   cadr(cons(z0, cons(z1, z2))) -> z1
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   p(s(z0)) -> z0
   p(0) -> 0
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   prod(z0) -> if(shorter(z0, 0), shorter(z0, s(0)), z0)
   if(true, z0, z1) -> s(0)
   if(false, z0, z1) -> if2(z0, z1)
   if2(true, z0) -> car(z0)
   if2(false, z0) -> prod(cons(times(car(z0), cadr(z0)), cddr(z0)))
Tuples:
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1), ISZERO(z0))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1), P(z0))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1), ISZERO(z0))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1), P(z0))
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, 0))
   PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0)))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CAR(z0))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CADR(z0))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), CDDR(z0))
S tuples:
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1), ISZERO(z0))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1), P(z0))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1), ISZERO(z0))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1), P(z0))
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, 0))
   PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0)))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CAR(z0))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)), CADR(z0))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), CDDR(z0))
K tuples:none
Defined Rule Symbols:   car_1, cddr_1, cadr_1, isZero_1, plus_2, ifplus_3, times_2, iftimes_3, p_1, shorter_2, prod_1, if_3, if2_2

Defined Pair Symbols:   PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, SHORTER_2, PROD_1, IF_3, IF2_2

Compound Symbols:   c7_2, c9_2, c10_2, c12_3, c17_1, c18_2, c19_2, c21_1, c23_3, c24_3, c25_2


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

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   car(cons(z0, z1)) -> z0
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   cadr(cons(z0, cons(z1, z2))) -> z1
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   p(s(z0)) -> z0
   p(0) -> 0
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   prod(z0) -> if(shorter(z0, 0), shorter(z0, s(0)), z0)
   if(true, z0, z1) -> s(0)
   if(false, z0, z1) -> if2(z0, z1)
   if2(true, z0) -> car(z0)
   if2(false, z0) -> prod(cons(times(car(z0), cadr(z0)), cddr(z0)))
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0)))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0)))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
K tuples:none
Defined Rule Symbols:   car_1, cddr_1, cadr_1, isZero_1, plus_2, ifplus_3, times_2, iftimes_3, p_1, shorter_2, prod_1, if_3, if2_2

Defined Pair Symbols:   SHORTER_2, PROD_1, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, IF2_2

Compound Symbols:   c17_1, c19_2, c21_1, c7_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1


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

(85) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   prod(z0) -> if(shorter(z0, 0), shorter(z0, s(0)), z0)
   if(true, z0, z1) -> s(0)
   if(false, z0, z1) -> if2(z0, z1)
   if2(true, z0) -> car(z0)
   if2(false, z0) -> prod(cons(times(car(z0), cadr(z0)), cddr(z0)))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0)))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0)))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
K tuples:none
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, PROD_1, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, IF2_2

Compound Symbols:   c17_1, c19_2, c21_1, c7_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1


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

(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PROD(z0) -> c19(IF(shorter(z0, 0), shorter(z0, s(0)), z0), SHORTER(z0, s(0))) by 
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil), SHORTER(nil, s(0)))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(true, shorter(nil, s(0)), nil), SHORTER(nil, s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil), SHORTER(nil, s(0)))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(true, shorter(nil, s(0)), nil), SHORTER(nil, s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil), SHORTER(nil, s(0)))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(true, shorter(nil, s(0)), nil), SHORTER(nil, s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
K tuples:none
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2


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

(89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PROD(nil) -> c19(IF(true, shorter(nil, s(0)), nil), SHORTER(nil, s(0)))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil), SHORTER(nil, s(0)))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil), SHORTER(nil, s(0)))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
K tuples:none
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2


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

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
K tuples:none
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1


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

(93) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1


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

(95) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PLUS(z0, z1) -> c7(IFPLUS(isZero(z0), z0, z1)) by 
   PLUS(0, x1) -> c7(IFPLUS(true, 0, x1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(0, x1) -> c7(IFPLUS(true, 0, x1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(0, x1) -> c7(IFPLUS(true, 0, x1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2, PLUS_2

Compound Symbols:   c17_1, c21_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1


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

(97) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PLUS(0, x1) -> c7(IFPLUS(true, 0, x1))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2, PLUS_2

Compound Symbols:   c17_1, c21_1, c9_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1


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

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPLUS(false, z0, z1) -> c9(PLUS(p(z0), z1)) by 
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   IFPLUS(false, 0, x1) -> c9(PLUS(0, x1))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   IFPLUS(false, 0, x1) -> c9(PLUS(0, x1))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   IFPLUS(false, 0, x1) -> c9(PLUS(0, x1))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2, PLUS_2, IFPLUS_3

Compound Symbols:   c17_1, c21_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1


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

(101) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IFPLUS(false, 0, x1) -> c9(PLUS(0, x1))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2, PLUS_2, IFPLUS_3

Compound Symbols:   c17_1, c21_1, c10_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1


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

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TIMES(z0, z1) -> c10(IFTIMES(isZero(z0), z0, z1)) by 
   TIMES(0, x1) -> c10(IFTIMES(true, 0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(0, x1) -> c10(IFTIMES(true, 0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(0, x1) -> c10(IFTIMES(true, 0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, IFTIMES_3, PROD_1, IF2_2, PLUS_2, IFPLUS_3, TIMES_2

Compound Symbols:   c17_1, c21_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1


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

(105) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   TIMES(0, x1) -> c10(IFTIMES(true, 0, x1))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, IFTIMES_3, PROD_1, IF2_2, PLUS_2, IFPLUS_3, TIMES_2

Compound Symbols:   c17_1, c21_1, c12_2, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1


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

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(p(z0), z1)), TIMES(p(z0), z1)) by 
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, times(0, x1)), TIMES(p(0), x1))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, times(0, x1)), TIMES(p(0), x1))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, times(0, x1)), TIMES(p(0), x1))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PROD_1, IF2_2, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2


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

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PROD_1, IF2_2, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1


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

(111) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PROD_1, IF2_2, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c18_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1


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

(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PROD(z0) -> c18(IF(shorter(z0, 0), shorter(z0, s(0)), z0)) by 
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(nil) -> c18(IF(true, shorter(nil, s(0)), nil))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(nil) -> c18(IF(true, shorter(nil, s(0)), nil))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(nil) -> c18(IF(true, shorter(nil, s(0)), nil))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1


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

(115) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PROD(nil) -> c18(IF(true, shorter(nil, s(0)), nil))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1


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

(117) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c23_2, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1


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

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, z0) -> c23(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) by 
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)), TIMES(car(nil), cadr(nil)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)), TIMES(car(nil), cadr(nil)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)), TIMES(car(nil), cadr(nil)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2


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

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1


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

(123) 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, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
We considered the (Usable) Rules:none
And the Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IF(x_1, x_2, x_3)) = x_3
   POL(IF2(x_1, x_2)) = x_2
   POL(IFPLUS(x_1, x_2, x_3)) = x_1 + x_2
   POL(IFTIMES(x_1, x_2, x_3)) = x_1
   POL(PLUS(x_1, x_2)) = 0
   POL(PROD(x_1)) = x_1
   POL(SHORTER(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c19(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c23(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cadr(x_1)) = [1]
   POL(car(x_1)) = 0
   POL(cddr(x_1)) = [1]
   POL(cons(x_1, x_2)) = 0
   POL(false) = 0
   POL(ifplus(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(iftimes(x_1, x_2, x_3)) = [1]
   POL(isZero(x_1)) = 0
   POL(nil) = [1]
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(shorter(x_1, x_2)) = 0
   POL(times(x_1, x_2)) = 0
   POL(true) = 0

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0)))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c24_2, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1


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

(125) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, z0) -> c24(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))), TIMES(car(z0), cadr(z0))) by 
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)), TIMES(car(nil), cadr(nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)), TIMES(car(nil), cadr(nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)), TIMES(car(nil), cadr(nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2


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

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1


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

(129) 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, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
We considered the (Usable) Rules:none
And the Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IF(x_1, x_2, x_3)) = x_3
   POL(IF2(x_1, x_2)) = x_2
   POL(IFPLUS(x_1, x_2, x_3)) = x_1 + x_2
   POL(IFTIMES(x_1, x_2, x_3)) = x_1
   POL(PLUS(x_1, x_2)) = 0
   POL(PROD(x_1)) = x_1
   POL(SHORTER(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c19(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c23(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c24(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cadr(x_1)) = [1]
   POL(car(x_1)) = 0
   POL(cddr(x_1)) = [1]
   POL(cons(x_1, x_2)) = 0
   POL(false) = 0
   POL(ifplus(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(iftimes(x_1, x_2, x_3)) = [1]
   POL(isZero(x_1)) = 0
   POL(nil) = [1]
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(shorter(x_1, x_2)) = 0
   POL(times(x_1, x_2)) = 0
   POL(true) = 0

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0))))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, IF2_2, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c25_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1


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

(131) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, z0) -> c25(PROD(cons(times(car(z0), cadr(z0)), cddr(z0)))) by 
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, IF2_2

Compound Symbols:   c17_1, c21_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1


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

(133) 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, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
We considered the (Usable) Rules:none
And the Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IF(x_1, x_2, x_3)) = x_3
   POL(IF2(x_1, x_2)) = x_2
   POL(IFPLUS(x_1, x_2, x_3)) = x_1 + x_2
   POL(IFTIMES(x_1, x_2, x_3)) = x_1
   POL(PLUS(x_1, x_2)) = 0
   POL(PROD(x_1)) = x_1
   POL(SHORTER(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c19(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c23(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c24(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cadr(x_1)) = [1]
   POL(car(x_1)) = 0
   POL(cddr(x_1)) = [1]
   POL(cons(x_1, x_2)) = 0
   POL(false) = 0
   POL(ifplus(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(iftimes(x_1, x_2, x_3)) = [1]
   POL(isZero(x_1)) = 0
   POL(nil) = [1]
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(shorter(x_1, x_2)) = 0
   POL(times(x_1, x_2)) = 0
   POL(true) = 0

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, IF2_2

Compound Symbols:   c17_1, c21_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1


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

(135) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PROD(cons(z0, z1)) -> c19(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) by 
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, IF2_2

Compound Symbols:   c17_1, c21_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1


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

(137) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
We considered the (Usable) Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   shorter(cons(z0, z1), 0) -> false
And the Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(IF(x_1, x_2, x_3)) = x_2
   POL(IF2(x_1, x_2)) = x_1
   POL(IFPLUS(x_1, x_2, x_3)) = 0
   POL(IFTIMES(x_1, x_2, x_3)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(PROD(x_1)) = [1]
   POL(SHORTER(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c19(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c23(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c24(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cadr(x_1)) = [1]
   POL(car(x_1)) = 0
   POL(cddr(x_1)) = [1] + x_1
   POL(cons(x_1, x_2)) = 0
   POL(false) = [1]
   POL(ifplus(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(iftimes(x_1, x_2, x_3)) = [1]
   POL(isZero(x_1)) = 0
   POL(nil) = 0
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1]
   POL(s(x_1)) = x_1
   POL(shorter(x_1, x_2)) = x_2
   POL(times(x_1, x_2)) = 0
   POL(true) = 0

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, IF2_2

Compound Symbols:   c17_1, c21_1, c19_2, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1


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

(139) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PROD(cons(z0, z1)) -> c19(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)), SHORTER(cons(z0, z1), s(0))) by 
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PROD_1, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, IF2_2

Compound Symbols:   c17_1, c21_1, c19_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2


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

(141) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PROD(nil) -> c19(IF(shorter(nil, 0), true, nil)) by 
   PROD(nil) -> c19(IF(true, true, nil))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   PROD(nil) -> c19(IF(true, true, nil))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
K tuples:
   PROD(nil) -> c19(IF(shorter(nil, 0), true, nil))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c19_1


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

(143) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PROD(nil) -> c19(IF(true, true, nil))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2


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

(145) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFTIMES(false, x0, z1) -> c12(PLUS(z1, iftimes(isZero(p(x0)), p(x0), z1)), TIMES(p(x0), z1)) by 
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)), TIMES(0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)), TIMES(0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)), TIMES(0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_1


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

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_1


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

(149) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_1


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

(151) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(z0, x1)), TIMES(p(s(z0)), x1)) by 
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1


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

(153) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1))) by 
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IFTIMES_3, PROD_1, IF2_2

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1


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

(155) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1)) by 
   IFTIMES(false, 0, x0) -> c(TIMES(0, x0))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   IFTIMES(false, 0, x0) -> c(TIMES(0, x0))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(0, x1)))
   IFTIMES(false, 0, x1) -> c(TIMES(p(0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, PROD_1, IF2_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1


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

(157) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IFTIMES(false, 0, x0) -> c(TIMES(0, x0))

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
K tuples:
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, PROD_1, IF2_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1


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

(159) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PROD(nil) -> c18(IF(shorter(nil, 0), true, nil)) by 
   PROD(nil) -> c18(IF(true, true, nil))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(nil) -> c18(IF(true, true, nil))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
K tuples:
   PROD(nil) -> c18(IF(shorter(nil, 0), true, nil))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, PROD_1, IF2_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1


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

(161) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PROD(nil) -> c18(IF(true, true, nil))

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

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, PROD_1, IF2_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1


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

(163) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PROD(cons(z0, z1)) -> c18(IF(shorter(cons(z0, z1), 0), shorter(z1, 0), cons(z0, z1))) by 
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, PROD_1, IF2_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1


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

(165) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
We considered the (Usable) Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   shorter(cons(z0, z1), 0) -> false
And the Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IF(x_1, x_2, x_3)) = x_1 + x_2
   POL(IF2(x_1, x_2)) = [1] + x_1
   POL(IFPLUS(x_1, x_2, x_3)) = x_2
   POL(IFTIMES(x_1, x_2, x_3)) = x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(PROD(x_1)) = [1] + x_1
   POL(SHORTER(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c23(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c24(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cadr(x_1)) = [1]
   POL(car(x_1)) = 0
   POL(cddr(x_1)) = [1]
   POL(cons(x_1, x_2)) = [1]
   POL(false) = [1]
   POL(ifplus(x_1, x_2, x_3)) = [1] + x_2 + x_3
   POL(iftimes(x_1, x_2, x_3)) = 0
   POL(isZero(x_1)) = 0
   POL(nil) = 0
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(shorter(x_1, x_2)) = [1]
   POL(times(x_1, x_2)) = [1]
   POL(true) = 0

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, PROD_1, IF2_2, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c18_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1


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

(167) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PROD(cons(z0, z1)) -> c18(IF(false, shorter(cons(z0, z1), s(0)), cons(z0, z1))) by 
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)
   isZero(0) -> true
   isZero(s(z0)) -> false
   p(s(z0)) -> z0
   p(0) -> 0
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   car(cons(z0, z1)) -> z0
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
Defined Rule Symbols:   shorter_2, isZero_1, p_1, times_2, iftimes_3, plus_2, ifplus_3, car_1, cadr_1, cddr_1

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1


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

(169) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   shorter(cons(z0, z1), s(z2)) -> shorter(z1, z2)

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1


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

(171) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) by 
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1


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

(173) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) by 
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1


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

(175) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) by 
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))

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

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1


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

(177) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) by 
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))

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

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c23_2, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1


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

(179) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, cons(z0, z1)) -> c23(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) by 
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(z0, z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))

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

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2


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

(181) 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, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
We considered the (Usable) Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
And the Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IF(x_1, x_2, x_3)) = x_2
   POL(IF2(x_1, x_2)) = x_1
   POL(IFPLUS(x_1, x_2, x_3)) = x_2
   POL(IFTIMES(x_1, x_2, x_3)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(PROD(x_1)) = [1]
   POL(SHORTER(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c23(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c24(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cadr(x_1)) = [1]
   POL(car(x_1)) = 0
   POL(cddr(x_1)) = [1] + x_1
   POL(cons(x_1, x_2)) = 0
   POL(false) = [1]
   POL(ifplus(x_1, x_2, x_3)) = [1] + x_2 + x_3
   POL(iftimes(x_1, x_2, x_3)) = 0
   POL(isZero(x_1)) = 0
   POL(nil) = 0
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(shorter(x_1, x_2)) = [1] + x_2
   POL(times(x_1, x_2)) = [1]
   POL(true) = 0

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

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c23_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2


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

(183) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil))) by 
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))

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

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1


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

(185) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) by 
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))

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

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1


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

(187) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) by 
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))

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

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0)))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1


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

(189) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))), TIMES(car(x0), cadr(x0))) by 
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))

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

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1


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

(191) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2))))) by 
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))

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

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1))))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c24_2, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1


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

(193) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, cons(z0, z1)) -> c24(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(car(cons(z0, z1)), cadr(cons(z0, z1)))) by 
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(z1, z2)))), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(z0, z1), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))

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

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2


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

(195) 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, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
We considered the (Usable) Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
And the Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IF(x_1, x_2, x_3)) = x_1 + x_2
   POL(IF2(x_1, x_2)) = [1] + x_1
   POL(IFPLUS(x_1, x_2, x_3)) = x_2
   POL(IFTIMES(x_1, x_2, x_3)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(PROD(x_1)) = [1] + x_1
   POL(SHORTER(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c23(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c24(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cadr(x_1)) = [1]
   POL(car(x_1)) = 0
   POL(cddr(x_1)) = [1]
   POL(cons(x_1, x_2)) = [1]
   POL(false) = [1]
   POL(ifplus(x_1, x_2, x_3)) = [1] + x_2 + x_3
   POL(iftimes(x_1, x_2, x_3)) = 0
   POL(isZero(x_1)) = 0
   POL(nil) = 0
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(shorter(x_1, x_2)) = [1] + x_2
   POL(times(x_1, x_2)) = [1]
   POL(true) = 0

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

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c24_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2


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

(197) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil))) by 
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))

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

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1


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

(199) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil))) by 
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))

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

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1


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

(201) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(car(cons(z0, nil)), cadr(cons(z0, nil))), nil))) by 
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))

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

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1


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

(203) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), z2))) by 
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))

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

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2))))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1


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

(205) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), cddr(cons(z0, cons(z1, z2)))))) by 
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))

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

(206)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1)))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1


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

(207) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, cons(z0, z1)) -> c25(PROD(cons(times(z0, cadr(cons(z0, z1))), cddr(cons(z0, z1))))) by 
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(z1, z2)))), z2)))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(z0, z1), cddr(cons(z0, cons(z1, z2))))))

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

(208)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1


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

(209) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PROD(cons(x0, cons(z0, z1))) -> c19(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))), SHORTER(cons(x0, cons(z0, z1)), s(0))) by 
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))

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

(210)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1


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

(211) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0))) by 
   PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0)))

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

(212)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1


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

(213) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), p(s(z0)), x1)), TIMES(z0, x1)) by 
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))

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

(214)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1


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

(215) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1))) by 
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0)))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0)))

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

(216)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0)))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c12_2, c_1, c18_1, c23_2, c23_1, c24_2, c24_1


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

(217) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, iftimes(isZero(z0), z0, z1)), TIMES(p(s(z0)), z1)) by 
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))

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

(218)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0)))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2


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

(219) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFTIMES(false, 0, z1) -> c(PLUS(z1, iftimes(isZero(0), 0, z1))) by 
   IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0)))

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

(220)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0)))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1))))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2, c_1


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

(221) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PROD(cons(x0, cons(z0, z1))) -> c18(IF(shorter(cons(x0, cons(z0, z1)), 0), false, cons(x0, cons(z0, z1)))) by 
   PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2))))

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

(222)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0)))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0)))
   PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
   PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2, c_1


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

(223) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil))) by 
   PROD(cons(z0, nil)) -> c18(IF(false, true, cons(z0, nil)))

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

(224)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0)))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0)))
   PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2))))
   PROD(cons(z0, nil)) -> c18(IF(false, true, cons(z0, nil)))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
   PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2, c_1


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

(225) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, cons(x0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil)))) by 
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))

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

(226)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0)))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0)))
   PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2))))
   PROD(cons(z0, nil)) -> c18(IF(false, true, cons(z0, nil)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
   PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2, c_1


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

(227) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil)))) by 
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))

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

(228)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0)))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0)))
   PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2))))
   PROD(cons(z0, nil)) -> c18(IF(false, true, cons(z0, nil)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
   PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2, c_1


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

(229) 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, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
We considered the (Usable) Rules:
   shorter(nil, z0) -> true
   shorter(cons(z0, z1), 0) -> false
And the Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0)))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0)))
   PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2))))
   PROD(cons(z0, nil)) -> c18(IF(false, true, cons(z0, nil)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(IF(x_1, x_2, x_3)) = [1] + x_2
   POL(IF2(x_1, x_2)) = [1] + x_1
   POL(IFPLUS(x_1, x_2, x_3)) = x_2
   POL(IFTIMES(x_1, x_2, x_3)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(PROD(x_1)) = [1] + x_1
   POL(SHORTER(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1)) = x_1
   POL(c23(x_1)) = x_1
   POL(c23(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c24(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cadr(x_1)) = [1]
   POL(car(x_1)) = 0
   POL(cddr(x_1)) = [1]
   POL(cons(x_1, x_2)) = [1]
   POL(false) = [1]
   POL(ifplus(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(iftimes(x_1, x_2, x_3)) = 0
   POL(isZero(x_1)) = 0
   POL(nil) = 0
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(shorter(x_1, x_2)) = x_2
   POL(times(x_1, x_2)) = [1]
   POL(true) = 0

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

(230)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0)))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0)))
   PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2))))
   PROD(cons(z0, nil)) -> c18(IF(false, true, cons(z0, nil)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
   PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2, c_1


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

(231) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, cons(x0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2))))) by 
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(x1, x2)))), z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))

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

(232)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> iftimes(isZero(z0), z0, z1)
   car(cons(z0, z1)) -> z0
   iftimes(true, z0, z1) -> 0
   iftimes(false, z0, z1) -> plus(z1, times(p(z0), z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   plus(z0, z1) -> ifplus(isZero(z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> 0
   ifplus(true, z0, z1) -> z1
   ifplus(false, z0, z1) -> s(plus(p(z0), z1))
   cadr(cons(z0, cons(z1, z2))) -> z1
   cddr(nil) -> nil
   cddr(cons(z0, nil)) -> nil
   cddr(cons(z0, cons(z1, z2))) -> z2
   shorter(cons(z0, z1), 0) -> false
   shorter(nil, z0) -> true
Tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c23(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, nil) -> c24(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(iftimes(isZero(car(nil)), car(nil), cadr(nil)), nil)))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   PROD(cons(z0, nil)) -> c19(IF(false, true, cons(z0, nil)), SHORTER(cons(z0, nil), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(p(0)), 0, x0)))
   IFTIMES(false, 0, x0) -> c12(PLUS(x0, iftimes(isZero(0), p(0), x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, 0, x0) -> c(PLUS(x0, iftimes(true, 0, x0)))
   PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2))))
   PROD(cons(z0, nil)) -> c18(IF(false, true, cons(z0, nil)))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(x1, x2)))), z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
S tuples:
   SHORTER(cons(z0, z1), s(z2)) -> c17(SHORTER(z1, z2))
   IF(false, z0, z1) -> c21(IF2(z0, z1))
   PLUS(s(z0), x1) -> c7(IFPLUS(false, s(z0), x1))
   IFPLUS(false, s(z0), x1) -> c9(PLUS(z0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF2(false, x0) -> c25(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   PROD(cons(z0, z1)) -> c19(IF(false, shorter(z1, 0), cons(z0, z1)), SHORTER(cons(z0, z1), s(0)))
   IFTIMES(false, x0, x1) -> c12(PLUS(x1, iftimes(isZero(p(x0)), p(x0), x1)))
   PROD(cons(z0, z1)) -> c18(IF(false, shorter(z1, 0), cons(z0, z1)))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c23(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)), TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
   IF2(false, cons(z0, nil)) -> c24(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(x0, cons(x1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)), TIMES(car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(z1, z2))) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))), cddr(cons(z0, cons(z1, z2))))), TIMES(car(cons(z0, cons(z1, z2))), z1))
   IF2(false, cons(z0, z1)) -> c24(PROD(cons(iftimes(isZero(car(cons(z0, z1))), car(cons(z0, z1)), cadr(cons(z0, z1))), cddr(cons(z0, z1)))), TIMES(z0, cadr(cons(z0, z1))))
   IF2(false, x0) -> c24(PROD(cons(iftimes(isZero(car(x0)), car(x0), cadr(x0)), cddr(x0))))
   IF2(false, cons(x0, cons(z1, x2))) -> c24(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))), TIMES(car(cons(x0, cons(z1, x2))), cadr(cons(x0, cons(z1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c24(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, x1)) -> c24(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))), TIMES(car(cons(z0, x1)), cadr(cons(z0, x1))))
   IF2(false, cons(x0, nil)) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, nil))), car(cons(x0, nil)), cadr(cons(x0, nil))), nil)))
   IF2(false, cons(z0, nil)) -> c25(PROD(cons(times(z0, cadr(cons(z0, nil))), nil)))
   IF2(false, cons(x0, cons(x1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(x1, x2)))), car(cons(x0, cons(x1, x2))), cadr(cons(x0, cons(x1, x2)))), x2)))
   IF2(false, cons(z0, cons(z1, z2))) -> c25(PROD(cons(times(car(cons(z0, cons(z1, z2))), z1), z2)))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, cadr(cons(z0, cons(x1, x2)))), x2)))
   IF2(false, cons(x0, cons(z1, x2))) -> c25(PROD(cons(iftimes(isZero(car(cons(x0, cons(z1, x2)))), car(cons(x0, cons(z1, x2))), z1), cddr(cons(x0, cons(z1, x2))))))
   IF2(false, cons(z0, cons(x1, x2))) -> c25(PROD(cons(times(z0, x1), cddr(cons(z0, cons(x1, x2))))))
   IF2(false, cons(z0, x1)) -> c25(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, x1))), cddr(cons(z0, x1)))))
   PROD(cons(z0, cons(x1, x2))) -> c19(IF(false, false, cons(z0, cons(x1, x2))), SHORTER(cons(z0, cons(x1, x2)), s(0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(p(s(z0))), z0, x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), p(s(z0)), x1)), TIMES(z0, x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, iftimes(isZero(z0), z0, x1)), TIMES(z0, x1))
   PROD(cons(z0, cons(x1, x2))) -> c18(IF(false, false, cons(z0, cons(x1, x2))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, nil))), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, nil)), cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, nil)) -> c23(PROD(cons(iftimes(isZero(z0), z0, cadr(cons(z0, nil))), nil)), TIMES(car(cons(z0, nil)), cadr(cons(z0, nil))))
   IF2(false, cons(z0, cons(z1, z2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(z1, z2)))), car(cons(z0, cons(z1, z2))), z1), z2)), TIMES(car(cons(z0, cons(z1, z2))), cadr(cons(z0, cons(z1, z2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(car(cons(z0, cons(x1, x2)))), z0, cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
   IF2(false, cons(z0, cons(x1, x2))) -> c23(PROD(cons(iftimes(isZero(z0), car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))), x2)), TIMES(car(cons(z0, cons(x1, x2))), cadr(cons(z0, cons(x1, x2)))))
K tuples:
   IF2(false, nil) -> c23(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c24(PROD(cons(times(car(nil), cadr(nil)), nil)))
   IF2(false, nil) -> c25(PROD(cons(times(car(nil), cadr(nil)), nil)))
   PROD(cons(x0, nil)) -> c19(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)), SHORTER(cons(x0, nil), s(0)))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, iftimes(isZero(p(0)), p(0), x1)))
   PROD(cons(x0, nil)) -> c18(IF(shorter(cons(x0, nil), 0), true, cons(x0, nil)))
   IF2(false, cons(x0, x1)) -> c23(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, x1)) -> c24(TIMES(car(cons(x0, x1)), cadr(cons(x0, x1))))
   IF2(false, cons(x0, nil)) -> c23(TIMES(car(cons(x0, nil)), cadr(cons(x0, nil))))
Defined Rule Symbols:   times_2, car_1, iftimes_3, isZero_1, plus_2, p_1, ifplus_3, cadr_1, cddr_1, shorter_2

Defined Pair Symbols:   SHORTER_2, IF_3, PLUS_2, IFPLUS_3, TIMES_2, IF2_2, PROD_1, IFTIMES_3

Compound Symbols:   c17_1, c21_1, c7_1, c9_1, c10_1, c25_1, c19_2, c12_1, c18_1, c23_2, c23_1, c24_2, c24_1, c12_2, c_1