KILLED
proof of input_4Kha5pcsGG.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 18 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 126 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 57 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 2 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 192 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 54 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 243 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 88 ms]
        (36) CpxRNTS
        (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 150 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
        (42) CpxRNTS
        (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 242 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 64 ms]
        (48) CpxRNTS
        (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (50) CpxRNTS
        (51) IntTrsBoundProof [UPPER BOUND(ID), 95 ms]
        (52) CpxRNTS
        (53) IntTrsBoundProof [UPPER BOUND(ID), 12 ms]
        (54) CpxRNTS
        (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (56) CpxRNTS
        (57) IntTrsBoundProof [UPPER BOUND(ID), 7999 ms]
        (58) CpxRNTS
        (59) IntTrsBoundProof [UPPER BOUND(ID), 2142 ms]
        (60) CpxRNTS
    (61) CompletionProof [UPPER BOUND(ID), 0 ms]
    (62) CpxTypedWeightedCompleteTrs
        (63) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (64) CpxRNTS
(65) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(66) CdtProblem
    (67) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (68) CdtProblem
    (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 40 ms]
    (92) CdtProblem
    (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2277 ms]
    (102) CdtProblem
    (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 28 ms]
    (114) CdtProblem
    (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 23 ms]
    (122) CdtProblem
    (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms]
    (126) CdtProblem
    (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 49 ms]
    (134) CdtProblem
    (135) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 60 ms]
    (136) CdtProblem
    (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 45 ms]
    (140) CdtProblem
    (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (146) CdtProblem
    (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (154) CdtProblem
    (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem
    (175) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (176) CdtProblem
    (177) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (178) CdtProblem
    (179) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (180) CdtProblem
    (181) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (182) CdtProblem
    (183) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (184) CdtProblem
    (185) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (186) CdtProblem
    (187) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (188) CdtProblem
    (189) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (190) CdtProblem
    (191) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (192) CdtProblem
    (193) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (194) CdtProblem
    (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (196) CdtProblem
    (197) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (198) CdtProblem
    (199) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (200) CdtProblem
    (201) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (202) CdtProblem
    (203) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (204) CdtProblem
    (205) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (206) CdtProblem
    (207) CdtUsableRulesProof [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) CdtUsableRulesProof [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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (222) CdtProblem
    (223) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (224) CdtProblem
    (225) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (226) CdtProblem
    (227) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (228) CdtProblem
    (229) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (230) CdtProblem
    (231) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (232) CdtProblem
    (233) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (234) CdtProblem
    (235) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 75 ms]
    (236) CdtProblem
    (237) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (238) 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:

   prod(xs) -> prodIter(xs, s(0))
   prodIter(xs, x) -> ifProd(isempty(xs), xs, x)
   ifProd(true, xs, x) -> x
   ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> timesIter(x, y, 0, 0)
   timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u)
   ifTimes(true, x, y, z, u) -> z
   ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u))
   isempty(nil) -> true
   isempty(cons(x, xs)) -> false
   head(nil) -> error
   head(cons(x, xs)) -> x
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> b
   a -> c

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:

   prod(xs) -> prodIter(xs, s(0'))
   prodIter(xs, x) -> ifProd(isempty(xs), xs, x)
   ifProd(true, xs, x) -> x
   ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs)))
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> timesIter(x, y, 0', 0')
   timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u)
   ifTimes(true, x, y, z, u) -> z
   ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u))
   isempty(nil) -> true
   isempty(cons(x, xs)) -> false
   head(nil) -> error
   head(cons(x, xs)) -> x
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> b
   a -> c

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:

   prod(xs) -> prodIter(xs, s(0))
   prodIter(xs, x) -> ifProd(isempty(xs), xs, x)
   ifProd(true, xs, x) -> x
   ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> timesIter(x, y, 0, 0)
   timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u)
   ifTimes(true, x, y, z, u) -> z
   ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u))
   isempty(nil) -> true
   isempty(cons(x, xs)) -> false
   head(nil) -> error
   head(cons(x, xs)) -> x
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> b
   a -> c

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:

   prod(xs) -> prodIter(xs, s(0)) [1]
   prodIter(xs, x) -> ifProd(isempty(xs), xs, x) [1]
   ifProd(true, xs, x) -> x [1]
   ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(x, y) -> timesIter(x, y, 0, 0) [1]
   timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) [1]
   ifTimes(true, x, y, z, u) -> z [1]
   ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) [1]
   isempty(nil) -> true [1]
   isempty(cons(x, xs)) -> false [1]
   head(nil) -> error [1]
   head(cons(x, xs)) -> x [1]
   tail(nil) -> nil [1]
   tail(cons(x, xs)) -> xs [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   a -> b [1]
   a -> c [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:

   prod(xs) -> prodIter(xs, s(0)) [1]
   prodIter(xs, x) -> ifProd(isempty(xs), xs, x) [1]
   ifProd(true, xs, x) -> x [1]
   ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(x, y) -> timesIter(x, y, 0, 0) [1]
   timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) [1]
   ifTimes(true, x, y, z, u) -> z [1]
   ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) [1]
   isempty(nil) -> true [1]
   isempty(cons(x, xs)) -> false [1]
   head(nil) -> error [1]
   head(cons(x, xs)) -> x [1]
   tail(nil) -> nil [1]
   tail(cons(x, xs)) -> xs [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   a -> b [1]
   a -> c [1]

The TRS has the following type information:
prod :: nil:cons -> 0:s:error
prodIter :: nil:cons -> 0:s:error -> 0:s:error
s :: 0:s:error -> 0:s:error
0 :: 0:s:error
ifProd :: true:false -> nil:cons -> 0:s:error -> 0:s:error
isempty :: nil:cons -> true:false
true :: true:false
false :: true:false
tail :: nil:cons -> nil:cons
times :: 0:s:error -> 0:s:error -> 0:s:error
head :: nil:cons -> 0:s:error
plus :: 0:s:error -> 0:s:error -> 0:s:error
timesIter :: 0:s:error -> 0:s:error -> 0:s:error -> 0:s:error -> 0:s:error
ifTimes :: true:false -> 0:s:error -> 0:s:error -> 0:s:error -> 0:s:error -> 0:s:error
ge :: 0:s:error -> 0:s:error -> true:false
nil :: nil:cons
cons :: 0:s:error -> nil:cons -> nil:cons
error :: 0:s:error
a :: b:c
b :: b:c
c :: b:c

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
   prodIter_2
   ifProd_3
   a

(c) The following functions are completely defined:

   tail_1
   times_2
   head_1
   isempty_1
   plus_2
   ge_2
   timesIter_4
   ifTimes_5

Due to the following rules being added:

   plus(v0, v1) -> null_plus [0]
   ge(v0, v1) -> null_ge [0]
   ifTimes(v0, v1, v2, v3, v4) -> null_ifTimes [0]

And the following fresh constants:    null_plus, null_ge, null_ifTimes

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

(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:

   prod(xs) -> prodIter(xs, s(0)) [1]
   prodIter(xs, x) -> ifProd(isempty(xs), xs, x) [1]
   ifProd(true, xs, x) -> x [1]
   ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(x, y) -> timesIter(x, y, 0, 0) [1]
   timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) [1]
   ifTimes(true, x, y, z, u) -> z [1]
   ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) [1]
   isempty(nil) -> true [1]
   isempty(cons(x, xs)) -> false [1]
   head(nil) -> error [1]
   head(cons(x, xs)) -> x [1]
   tail(nil) -> nil [1]
   tail(cons(x, xs)) -> xs [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   a -> b [1]
   a -> c [1]
   plus(v0, v1) -> null_plus [0]
   ge(v0, v1) -> null_ge [0]
   ifTimes(v0, v1, v2, v3, v4) -> null_ifTimes [0]

The TRS has the following type information:
prod :: nil:cons -> 0:s:error:null_plus:null_ifTimes
prodIter :: nil:cons -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes
s :: 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes
0 :: 0:s:error:null_plus:null_ifTimes
ifProd :: true:false:null_ge -> nil:cons -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes
isempty :: nil:cons -> true:false:null_ge
true :: true:false:null_ge
false :: true:false:null_ge
tail :: nil:cons -> nil:cons
times :: 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes
head :: nil:cons -> 0:s:error:null_plus:null_ifTimes
plus :: 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes
timesIter :: 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes
ifTimes :: true:false:null_ge -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes
ge :: 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> true:false:null_ge
nil :: nil:cons
cons :: 0:s:error:null_plus:null_ifTimes -> nil:cons -> nil:cons
error :: 0:s:error:null_plus:null_ifTimes
a :: b:c
b :: b:c
c :: b:c
null_plus :: 0:s:error:null_plus:null_ifTimes
null_ge :: true:false:null_ge
null_ifTimes :: 0:s:error:null_plus:null_ifTimes

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:

   prod(xs) -> prodIter(xs, s(0)) [1]
   prodIter(nil, x) -> ifProd(true, nil, x) [2]
   prodIter(cons(x', xs'), x) -> ifProd(false, cons(x', xs'), x) [2]
   ifProd(true, xs, x) -> x [1]
   ifProd(false, nil, x) -> prodIter(nil, times(x, error)) [3]
   ifProd(false, cons(x'', xs''), x) -> prodIter(xs'', times(x, x'')) [3]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(x, y) -> timesIter(x, y, 0, 0) [1]
   timesIter(0, y, z, u) -> ifTimes(true, 0, y, z, u) [2]
   timesIter(s(y'), y, z, 0) -> ifTimes(false, s(y'), y, z, 0) [2]
   timesIter(s(y''), y, z, s(x1)) -> ifTimes(ge(x1, y''), s(y''), y, z, s(x1)) [2]
   timesIter(x, y, z, u) -> ifTimes(null_ge, x, y, z, u) [1]
   ifTimes(true, x, y, z, u) -> z [1]
   ifTimes(false, x, 0, z, u) -> timesIter(x, 0, z, s(u)) [2]
   ifTimes(false, x, s(x2), z, u) -> timesIter(x, s(x2), s(plus(x2, z)), s(u)) [2]
   ifTimes(false, x, y, z, u) -> timesIter(x, y, null_plus, s(u)) [1]
   isempty(nil) -> true [1]
   isempty(cons(x, xs)) -> false [1]
   head(nil) -> error [1]
   head(cons(x, xs)) -> x [1]
   tail(nil) -> nil [1]
   tail(cons(x, xs)) -> xs [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   a -> b [1]
   a -> c [1]
   plus(v0, v1) -> null_plus [0]
   ge(v0, v1) -> null_ge [0]
   ifTimes(v0, v1, v2, v3, v4) -> null_ifTimes [0]

The TRS has the following type information:
prod :: nil:cons -> 0:s:error:null_plus:null_ifTimes
prodIter :: nil:cons -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes
s :: 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes
0 :: 0:s:error:null_plus:null_ifTimes
ifProd :: true:false:null_ge -> nil:cons -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes
isempty :: nil:cons -> true:false:null_ge
true :: true:false:null_ge
false :: true:false:null_ge
tail :: nil:cons -> nil:cons
times :: 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes
head :: nil:cons -> 0:s:error:null_plus:null_ifTimes
plus :: 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes
timesIter :: 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes
ifTimes :: true:false:null_ge -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes
ge :: 0:s:error:null_plus:null_ifTimes -> 0:s:error:null_plus:null_ifTimes -> true:false:null_ge
nil :: nil:cons
cons :: 0:s:error:null_plus:null_ifTimes -> nil:cons -> nil:cons
error :: 0:s:error:null_plus:null_ifTimes
a :: b:c
b :: b:c
c :: b:c
null_plus :: 0:s:error:null_plus:null_ifTimes
null_ge :: true:false:null_ge
null_ifTimes :: 0:s:error:null_plus:null_ifTimes

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

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

   0 => 0
   true => 2
   false => 1
   nil => 0
   error => 1
   b => 0
   c => 1
   null_plus => 0
   null_ge => 0
   null_ifTimes => 0

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = x, x >= 0
   ge(z', z'') -{ 1 }-> 1 :|: y >= 0, z'' = 1 + y, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> x :|: xs >= 0, z' = 2, x >= 0, z'' = xs, z1 = x
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(x, x'')) :|: xs'' >= 0, x >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs'', z1 = x
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(x, 1)) :|: z'' = 0, x >= 0, z' = 1, z1 = x
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x, z3 = u, u >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(x, y, 0, 1 + u) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1, z3 = u, u >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(x, 0, z, 1 + u) :|: z1 = 0, z >= 0, z2 = z, x >= 0, z'' = x, z' = 1, z3 = u, u >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(x, 1 + x2, 1 + plus(x2, z), 1 + u) :|: z >= 0, z2 = z, x >= 0, z'' = x, z' = 1, z3 = u, x2 >= 0, z1 = 1 + x2, u >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, v4 >= 0, z1 = v2, v1 >= 0, z'' = v1, z3 = v4, v2 >= 0, v3 >= 0, z' = v0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   prod(z') -{ 1 }-> prodIter(xs, 1 + 0) :|: xs >= 0, z' = xs
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, x) :|: x >= 0, z'' = x, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', x) :|: x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = x
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(x, y, 0, 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(ge(x1, y''), 1 + y'', y, z, 1 + x1) :|: z' = 1 + y'', z1 = z, z >= 0, x1 >= 0, z'' = y, y >= 0, z2 = 1 + x1, y'' >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, y, z, u) :|: z1 = z, z2 = u, z >= 0, z'' = y, y >= 0, z' = 0, u >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + y', y, z, 0) :|: z1 = z, z >= 0, z'' = y, z2 = 0, y >= 0, y' >= 0, z' = 1 + y'
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, x, y, z, u) :|: z1 = z, z2 = u, z >= 0, z' = x, z'' = y, x >= 0, y >= 0, u >= 0


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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(ge(z2 - 1, z' - 1), 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0


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

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

   { tail }
   { isempty }
   { ge }
   { head }
   { plus }
   { a }
   { timesIter, ifTimes }
   { times }
   { prodIter, ifProd }
   { prod }

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(ge(z2 - 1, z' - 1), 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {tail}, {isempty}, {ge}, {head}, {plus}, {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(ge(z2 - 1, z' - 1), 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {tail}, {isempty}, {ge}, {head}, {plus}, {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(ge(z2 - 1, z' - 1), 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {tail}, {isempty}, {ge}, {head}, {plus}, {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: ?, size: O(n^1) [z']

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(ge(z2 - 1, z' - 1), 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {isempty}, {ge}, {head}, {plus}, {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(ge(z2 - 1, z' - 1), 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {isempty}, {ge}, {head}, {plus}, {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: isempty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(ge(z2 - 1, z' - 1), 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {isempty}, {ge}, {head}, {plus}, {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: ?, size: O(1) [2]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: isempty
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:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(ge(z2 - 1, z' - 1), 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {ge}, {head}, {plus}, {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(ge(z2 - 1, z' - 1), 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {ge}, {head}, {plus}, {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: ge
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(ge(z2 - 1, z' - 1), 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {ge}, {head}, {plus}, {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  ge: runtime: ?, size: O(1) [2]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(ge(z2 - 1, z' - 1), 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {head}, {plus}, {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 3 + z2 }-> ifTimes(s, 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: s >= 0, s <= 2, z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {head}, {plus}, {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 3 + z2 }-> ifTimes(s, 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: s >= 0, s <= 2, z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {head}, {plus}, {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: ?, size: O(n^1) [1 + z']

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 3 + z2 }-> ifTimes(s, 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: s >= 0, s <= 2, z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {plus}, {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 3 + z2 }-> ifTimes(s, 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: s >= 0, s <= 2, z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {plus}, {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']

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

(45) 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''

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 3 + z2 }-> ifTimes(s, 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: s >= 0, s <= 2, z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {plus}, {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  plus: runtime: ?, size: O(n^1) [z' + z'']

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

(47) 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: 1 + z'

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 1 + (z1 - 1), 1 + plus(z1 - 1, z2), 1 + z3) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 3 + z2 }-> ifTimes(s, 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: s >= 0, s <= 2, z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 + z1 }-> timesIter(z'', 1 + (z1 - 1), 1 + s1, 1 + z3) :|: s1 >= 0, s1 <= z1 - 1 + z2, z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 3 + z2 }-> ifTimes(s, 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: s >= 0, s <= 2, z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 + z1 }-> timesIter(z'', 1 + (z1 - 1), 1 + s1, 1 + z3) :|: s1 >= 0, s1 <= z1 - 1 + z2, z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 3 + z2 }-> ifTimes(s, 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: s >= 0, s <= 2, z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {a}, {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  a: runtime: ?, size: O(1) [1]

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 + z1 }-> timesIter(z'', 1 + (z1 - 1), 1 + s1, 1 + z3) :|: s1 >= 0, s1 <= z1 - 1 + z2, z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 3 + z2 }-> ifTimes(s, 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: s >= 0, s <= 2, z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  a: runtime: O(1) [1], size: O(1) [1]

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 + z1 }-> timesIter(z'', 1 + (z1 - 1), 1 + s1, 1 + z3) :|: s1 >= 0, s1 <= z1 - 1 + z2, z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 3 + z2 }-> ifTimes(s, 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: s >= 0, s <= 2, z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  a: runtime: O(1) [1], size: O(1) [1]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 + z1 }-> timesIter(z'', 1 + (z1 - 1), 1 + s1, 1 + z3) :|: s1 >= 0, s1 <= z1 - 1 + z2, z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 3 + z2 }-> ifTimes(s, 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: s >= 0, s <= 2, z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  a: runtime: O(1) [1], size: O(1) [1]
  timesIter: runtime: ?, size: INF
  ifTimes: runtime: ?, size: INF

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0
   ifProd(z', z'', z1) -{ 3 }-> prodIter(xs'', times(z1, x'')) :|: xs'' >= 0, z1 >= 0, z' = 1, x'' >= 0, z'' = 1 + x'' + xs''
   ifProd(z', z'', z1) -{ 3 }-> prodIter(0, times(z1, 1)) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(z'', z1, 0, 1 + z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 }-> timesIter(z'', 0, z2, 1 + z3) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 2 + z1 }-> timesIter(z'', 1 + (z1 - 1), 1 + s1, 1 + z3) :|: s1 >= 0, s1 <= z1 - 1 + z2, z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   prod(z') -{ 1 }-> prodIter(z', 1 + 0) :|: z' >= 0
   prodIter(z', z'') -{ 2 }-> ifProd(2, 0, z'') :|: z'' >= 0, z' = 0
   prodIter(z', z'') -{ 2 }-> ifProd(1, 1 + x' + xs', z'') :|: x' >= 0, xs' >= 0, z'' >= 0, z' = 1 + x' + xs'
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(z', z'', 0, 0) :|: z' >= 0, z'' >= 0
   timesIter(z', z'', z1, z2) -{ 3 + z2 }-> ifTimes(s, 1 + (z' - 1), z'', z1, 1 + (z2 - 1)) :|: s >= 0, s <= 2, z1 >= 0, z2 - 1 >= 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(2, 0, z'', z1, z2) :|: z1 >= 0, z'' >= 0, z' = 0, z2 >= 0
   timesIter(z', z'', z1, z2) -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(0, z', z'', z1, z2) :|: z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0

Function symbols to be analyzed: {timesIter,ifTimes}, {times}, {prodIter,ifProd}, {prod}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  a: runtime: O(1) [1], size: O(1) [1]
  timesIter: runtime: INF, size: INF
  ifTimes: runtime: ?, size: INF

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

(61) 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:

   plus(v0, v1) -> null_plus [0]
   ge(v0, v1) -> null_ge [0]
   ifProd(v0, v1, v2) -> null_ifProd [0]
   ifTimes(v0, v1, v2, v3, v4) -> null_ifTimes [0]

And the following fresh constants:    null_plus, null_ge, null_ifProd, null_ifTimes

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

(62)
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:

   prod(xs) -> prodIter(xs, s(0)) [1]
   prodIter(xs, x) -> ifProd(isempty(xs), xs, x) [1]
   ifProd(true, xs, x) -> x [1]
   ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(x, y) -> timesIter(x, y, 0, 0) [1]
   timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) [1]
   ifTimes(true, x, y, z, u) -> z [1]
   ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) [1]
   isempty(nil) -> true [1]
   isempty(cons(x, xs)) -> false [1]
   head(nil) -> error [1]
   head(cons(x, xs)) -> x [1]
   tail(nil) -> nil [1]
   tail(cons(x, xs)) -> xs [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   a -> b [1]
   a -> c [1]
   plus(v0, v1) -> null_plus [0]
   ge(v0, v1) -> null_ge [0]
   ifProd(v0, v1, v2) -> null_ifProd [0]
   ifTimes(v0, v1, v2, v3, v4) -> null_ifTimes [0]

The TRS has the following type information:
prod :: nil:cons -> 0:s:error:null_plus:null_ifProd:null_ifTimes
prodIter :: nil:cons -> 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes
s :: 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes
0 :: 0:s:error:null_plus:null_ifProd:null_ifTimes
ifProd :: true:false:null_ge -> nil:cons -> 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes
isempty :: nil:cons -> true:false:null_ge
true :: true:false:null_ge
false :: true:false:null_ge
tail :: nil:cons -> nil:cons
times :: 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes
head :: nil:cons -> 0:s:error:null_plus:null_ifProd:null_ifTimes
plus :: 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes
timesIter :: 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes
ifTimes :: true:false:null_ge -> 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes
ge :: 0:s:error:null_plus:null_ifProd:null_ifTimes -> 0:s:error:null_plus:null_ifProd:null_ifTimes -> true:false:null_ge
nil :: nil:cons
cons :: 0:s:error:null_plus:null_ifProd:null_ifTimes -> nil:cons -> nil:cons
error :: 0:s:error:null_plus:null_ifProd:null_ifTimes
a :: b:c
b :: b:c
c :: b:c
null_plus :: 0:s:error:null_plus:null_ifProd:null_ifTimes
null_ge :: true:false:null_ge
null_ifProd :: 0:s:error:null_plus:null_ifProd:null_ifTimes
null_ifTimes :: 0:s:error:null_plus:null_ifProd:null_ifTimes

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

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

   0 => 0
   true => 2
   false => 1
   nil => 0
   error => 1
   b => 0
   c => 1
   null_plus => 0
   null_ge => 0
   null_ifProd => 0
   null_ifTimes => 0

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = x, x >= 0
   ge(z', z'') -{ 1 }-> 1 :|: y >= 0, z'' = 1 + y, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifProd(z', z'', z1) -{ 1 }-> x :|: xs >= 0, z' = 2, x >= 0, z'' = xs, z1 = x
   ifProd(z', z'', z1) -{ 1 }-> prodIter(tail(xs), times(x, head(xs))) :|: xs >= 0, x >= 0, z'' = xs, z' = 1, z1 = x
   ifProd(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> z :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x, z3 = u, u >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 1 }-> timesIter(x, y, plus(y, z), 1 + u) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1, z3 = u, u >= 0
   ifTimes(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, v4 >= 0, z1 = v2, v1 >= 0, z'' = v1, z3 = v4, v2 >= 0, v3 >= 0, z' = v0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   prod(z') -{ 1 }-> prodIter(xs, 1 + 0) :|: xs >= 0, z' = xs
   prodIter(z', z'') -{ 1 }-> ifProd(isempty(xs), xs, x) :|: xs >= 0, x >= 0, z'' = x, z' = xs
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 1 }-> timesIter(x, y, 0, 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   timesIter(z', z'', z1, z2) -{ 1 }-> ifTimes(ge(u, x), x, y, z, u) :|: z1 = z, z2 = u, z >= 0, z' = x, z'' = y, x >= 0, y >= 0, u >= 0

Only complete derivations are relevant for the runtime complexity.

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

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prod(z0) -> prodIter(z0, s(0))
   prodIter(z0, z1) -> ifProd(isempty(z0), z0, z1)
   ifProd(true, z0, z1) -> z1
   ifProd(false, z0, z1) -> prodIter(tail(z0), times(z1, head(z0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> b
   a -> c
Tuples:
   PROD(z0) -> c1(PRODITER(z0, s(0)))
   PRODITER(z0, z1) -> c2(IFPROD(isempty(z0), z0, z1), ISEMPTY(z0))
   IFPROD(true, z0, z1) -> c3
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))), TAIL(z0))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)), HEAD(z0))
   PLUS(0, z0) -> c6
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   TIMESITER(z0, z1, z2, z3) -> c9(IFTIMES(ge(z3, z0), z0, z1, z2, z3), GE(z3, z0))
   IFTIMES(true, z0, z1, z2, z3) -> c10
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   ISEMPTY(nil) -> c12
   ISEMPTY(cons(z0, z1)) -> c13
   HEAD(nil) -> c14
   HEAD(cons(z0, z1)) -> c15
   TAIL(nil) -> c16
   TAIL(cons(z0, z1)) -> c17
   GE(z0, 0) -> c18
   GE(0, s(z0)) -> c19
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   A -> c21
   A -> c22
S tuples:
   PROD(z0) -> c1(PRODITER(z0, s(0)))
   PRODITER(z0, z1) -> c2(IFPROD(isempty(z0), z0, z1), ISEMPTY(z0))
   IFPROD(true, z0, z1) -> c3
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))), TAIL(z0))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)), HEAD(z0))
   PLUS(0, z0) -> c6
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   TIMESITER(z0, z1, z2, z3) -> c9(IFTIMES(ge(z3, z0), z0, z1, z2, z3), GE(z3, z0))
   IFTIMES(true, z0, z1, z2, z3) -> c10
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   ISEMPTY(nil) -> c12
   ISEMPTY(cons(z0, z1)) -> c13
   HEAD(nil) -> c14
   HEAD(cons(z0, z1)) -> c15
   TAIL(nil) -> c16
   TAIL(cons(z0, z1)) -> c17
   GE(z0, 0) -> c18
   GE(0, s(z0)) -> c19
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   A -> c21
   A -> c22
K tuples:none
Defined Rule Symbols:   prod_1, prodIter_2, ifProd_3, plus_2, times_2, timesIter_4, ifTimes_5, isempty_1, head_1, tail_1, ge_2, a

Defined Pair Symbols:   PROD_1, PRODITER_2, IFPROD_3, PLUS_2, TIMES_2, TIMESITER_4, IFTIMES_5, ISEMPTY_1, HEAD_1, TAIL_1, GE_2, A

Compound Symbols:   c1_1, c2_2, c3, c4_2, c5_3, c6, c7_1, c8_1, c9_2, c10, c11_2, c12, c13, c14, c15, c16, c17, c18, c19, c20_1, c21, c22


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

(67) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   PROD(z0) -> c1(PRODITER(z0, s(0)))
Removed 13 trailing nodes:
   TAIL(nil) -> c16
   TAIL(cons(z0, z1)) -> c17
   ISEMPTY(nil) -> c12
   A -> c22
   GE(z0, 0) -> c18
   PLUS(0, z0) -> c6
   ISEMPTY(cons(z0, z1)) -> c13
   A -> c21
   IFPROD(true, z0, z1) -> c3
   HEAD(cons(z0, z1)) -> c15
   GE(0, s(z0)) -> c19
   HEAD(nil) -> c14
   IFTIMES(true, z0, z1, z2, z3) -> c10

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prod(z0) -> prodIter(z0, s(0))
   prodIter(z0, z1) -> ifProd(isempty(z0), z0, z1)
   ifProd(true, z0, z1) -> z1
   ifProd(false, z0, z1) -> prodIter(tail(z0), times(z1, head(z0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> b
   a -> c
Tuples:
   PRODITER(z0, z1) -> c2(IFPROD(isempty(z0), z0, z1), ISEMPTY(z0))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))), TAIL(z0))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)), HEAD(z0))
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   TIMESITER(z0, z1, z2, z3) -> c9(IFTIMES(ge(z3, z0), z0, z1, z2, z3), GE(z3, z0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
S tuples:
   PRODITER(z0, z1) -> c2(IFPROD(isempty(z0), z0, z1), ISEMPTY(z0))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))), TAIL(z0))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)), HEAD(z0))
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   TIMESITER(z0, z1, z2, z3) -> c9(IFTIMES(ge(z3, z0), z0, z1, z2, z3), GE(z3, z0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   prod_1, prodIter_2, ifProd_3, plus_2, times_2, timesIter_4, ifTimes_5, isempty_1, head_1, tail_1, ge_2, a

Defined Pair Symbols:   PRODITER_2, IFPROD_3, PLUS_2, TIMES_2, TIMESITER_4, IFTIMES_5, GE_2

Compound Symbols:   c2_2, c4_2, c5_3, c7_1, c8_1, c9_2, c11_2, c20_1


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

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prod(z0) -> prodIter(z0, s(0))
   prodIter(z0, z1) -> ifProd(isempty(z0), z0, z1)
   ifProd(true, z0, z1) -> z1
   ifProd(false, z0, z1) -> prodIter(tail(z0), times(z1, head(z0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> b
   a -> c
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   TIMESITER(z0, z1, z2, z3) -> c9(IFTIMES(ge(z3, z0), z0, z1, z2, z3), GE(z3, z0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(z0, z1) -> c2(IFPROD(isempty(z0), z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   TIMESITER(z0, z1, z2, z3) -> c9(IFTIMES(ge(z3, z0), z0, z1, z2, z3), GE(z3, z0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(z0, z1) -> c2(IFPROD(isempty(z0), z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
K tuples:none
Defined Rule Symbols:   prod_1, prodIter_2, ifProd_3, plus_2, times_2, timesIter_4, ifTimes_5, isempty_1, head_1, tail_1, ge_2, a

Defined Pair Symbols:   PLUS_2, TIMES_2, TIMESITER_4, IFTIMES_5, GE_2, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c9_2, c11_2, c20_1, c2_1, c4_1, c5_2


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

(71) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   prod(z0) -> prodIter(z0, s(0))
   prodIter(z0, z1) -> ifProd(isempty(z0), z0, z1)
   ifProd(true, z0, z1) -> z1
   ifProd(false, z0, z1) -> prodIter(tail(z0), times(z1, head(z0)))
   a -> b
   a -> c

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   TIMESITER(z0, z1, z2, z3) -> c9(IFTIMES(ge(z3, z0), z0, z1, z2, z3), GE(z3, z0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(z0, z1) -> c2(IFPROD(isempty(z0), z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   TIMESITER(z0, z1, z2, z3) -> c9(IFTIMES(ge(z3, z0), z0, z1, z2, z3), GE(z3, z0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(z0, z1) -> c2(IFPROD(isempty(z0), z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2, isempty_1, tail_1, times_2, head_1, timesIter_4, ifTimes_5

Defined Pair Symbols:   PLUS_2, TIMES_2, TIMESITER_4, IFTIMES_5, GE_2, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c9_2, c11_2, c20_1, c2_1, c4_1, c5_2


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

(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TIMESITER(z0, z1, z2, z3) -> c9(IFTIMES(ge(z3, z0), z0, z1, z2, z3), GE(z3, z0)) by 
   TIMESITER(0, x1, x2, z0) -> c9(IFTIMES(true, 0, x1, x2, z0), GE(z0, 0))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0), GE(0, s(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(z0, z1) -> c2(IFPROD(isempty(z0), z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(0, x1, x2, z0) -> c9(IFTIMES(true, 0, x1, x2, z0), GE(z0, 0))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0), GE(0, s(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(z0, z1) -> c2(IFPROD(isempty(z0), z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(0, x1, x2, z0) -> c9(IFTIMES(true, 0, x1, x2, z0), GE(z0, 0))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0), GE(0, s(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2, isempty_1, tail_1, times_2, head_1, timesIter_4, ifTimes_5

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c2_1, c4_1, c5_2, c9_2


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

(75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   TIMESITER(0, x1, x2, z0) -> c9(IFTIMES(true, 0, x1, x2, z0), GE(z0, 0))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(z0, z1) -> c2(IFPROD(isempty(z0), z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0), GE(0, s(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(z0, z1) -> c2(IFPROD(isempty(z0), z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0), GE(0, s(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2, isempty_1, tail_1, times_2, head_1, timesIter_4, ifTimes_5

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c2_1, c4_1, c5_2, c9_2


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

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(z0, z1) -> c2(IFPROD(isempty(z0), z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(z0, z1) -> c2(IFPROD(isempty(z0), z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2, isempty_1, tail_1, times_2, head_1, timesIter_4, ifTimes_5

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c2_1, c4_1, c5_2, c9_2, c9_1


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PRODITER(z0, z1) -> c2(IFPROD(isempty(z0), z0, z1)) by 
   PRODITER(nil, x1) -> c2(IFPROD(true, nil, x1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(nil, x1) -> c2(IFPROD(true, nil, x1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(nil, x1) -> c2(IFPROD(true, nil, x1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2, isempty_1, tail_1, times_2, head_1, timesIter_4, ifTimes_5

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, IFPROD_3, TIMESITER_4, PRODITER_2

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c4_1, c5_2, c9_2, c9_1, c2_1


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

(81) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PRODITER(nil, x1) -> c2(IFPROD(true, nil, x1))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2, isempty_1, tail_1, times_2, head_1, timesIter_4, ifTimes_5

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, IFPROD_3, TIMESITER_4, PRODITER_2

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c4_1, c5_2, c9_2, c9_1, c2_1


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

(83) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0))))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
K tuples:none
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, IFPROD_3, TIMESITER_4, PRODITER_2

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c4_1, c5_2, c9_2, c9_1, c2_1


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

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, z0, z1) -> c4(PRODITER(tail(z0), times(z1, head(z0)))) by 
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(nil, times(x1, head(nil))))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(nil, times(x1, head(nil))))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(nil, times(x1, head(nil))))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
K tuples:none
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, IFPROD_3, TIMESITER_4, PRODITER_2

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c5_2, c9_2, c9_1, c2_1, c4_1


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

(87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IFPROD(false, nil, x1) -> c4(PRODITER(nil, times(x1, head(nil))))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
K tuples:none
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, IFPROD_3, TIMESITER_4, PRODITER_2

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c5_2, c9_2, c9_1, c2_1, c4_1


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

(89) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
K tuples:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, IFPROD_3, TIMESITER_4, PRODITER_2

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c5_2, c9_2, c9_1, c2_1, c4_1


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

(91) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
We considered the (Usable) Rules:
   tail(cons(z0, z1)) -> z1
   tail(nil) -> nil
And the Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IFPROD(x_1, x_2, x_3)) = x_2
   POL(IFTIMES(x_1, x_2, x_3, x_4, x_5)) = x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(PRODITER(x_1, x_2)) = x_1
   POL(TIMES(x_1, x_2)) = 0
   POL(TIMESITER(x_1, x_2, x_3, x_4)) = x_4
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(error) = [1]
   POL(false) = [1]
   POL(ge(x_1, x_2)) = [1] + x_1 + x_2
   POL(head(x_1)) = [1]
   POL(ifTimes(x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(nil) = 0
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(times(x_1, x_2)) = [1] + x_1
   POL(timesIter(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(true) = [1]

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0)))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
K tuples:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, IFPROD_3, TIMESITER_4, PRODITER_2

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c5_2, c9_2, c9_1, c2_1, c4_1


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

(93) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, z0, z1) -> c5(PRODITER(tail(z0), times(z1, head(z0))), TIMES(z1, head(z0))) by 
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, nil, x1) -> c5(PRODITER(tail(nil), times(x1, error)), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(PRODITER(nil, times(x1, head(nil))), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, nil, x1) -> c5(PRODITER(tail(nil), times(x1, error)), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(PRODITER(nil, times(x1, head(nil))), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, nil, x1) -> c5(PRODITER(tail(nil), times(x1, error)), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(PRODITER(nil, times(x1, head(nil))), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
K tuples:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_2, c9_1, c2_1, c4_1, c5_2


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

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, nil, x1) -> c5(PRODITER(tail(nil), times(x1, error)), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, nil, x1) -> c5(PRODITER(tail(nil), times(x1, error)), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
K tuples:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_2, c9_1, c2_1, c4_1, c5_2, c5_1


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

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
K tuples:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_2, c9_1, c2_1, c4_1, c5_2, c5_1, c_1


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

(99) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
K tuples:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_2, c9_1, c2_1, c4_1, c5_2, c5_1, c_1


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

(101) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
We considered the (Usable) Rules:
   tail(cons(z0, z1)) -> z1
   tail(nil) -> nil
And the Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IFPROD(x_1, x_2, x_3)) = [2]x_1 + [2]x_1*x_3 + [2]x_1^2 + [2]x_2^2
   POL(IFTIMES(x_1, x_2, x_3, x_4, x_5)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(PRODITER(x_1, x_2)) = [2]x_1^2
   POL(TIMES(x_1, x_2)) = 0
   POL(TIMESITER(x_1, x_2, x_3, x_4)) = 0
   POL(c(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = [2] + x_2
   POL(error) = 0
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(head(x_1)) = 0
   POL(ifTimes(x_1, x_2, x_3, x_4, x_5)) = [1] + x_2 + x_3 + x_4 + x_5 + x_5^2 + x_4*x_5 + x_3*x_5 + x_2*x_5 + x_4^2 + x_3*x_4 + x_2*x_4 + x_2^2 + x_2*x_3 + x_3^2
   POL(nil) = 0
   POL(plus(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(times(x_1, x_2)) = 0
   POL(timesIter(x_1, x_2, x_3, x_4)) = 0
   POL(true) = [2]

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1)))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
K tuples:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_2, c9_1, c2_1, c4_1, c5_2, c5_1, c_1


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

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TIMESITER(s(z1), x1, x2, s(z0)) -> c9(IFTIMES(ge(z0, z1), s(z1), x1, x2, s(z0)), GE(s(z0), s(z1))) by 
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(IFTIMES(true, s(0), x1, x2, s(z0)), GE(s(z0), s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(IFTIMES(true, s(0), x1, x2, s(z0)), GE(s(z0), s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(IFTIMES(true, s(0), x1, x2, s(z0)), GE(s(z0), s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
K tuples:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c4_1, c5_2, c5_1, c_1, c9_2


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

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
K tuples:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c4_1, c5_2, c5_1, c_1, c9_2


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

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0))) by 
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(nil, timesIter(x1, head(nil), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(nil, timesIter(x1, head(nil), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(nil, timesIter(x1, head(nil), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
K tuples:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c4_1, c5_2, c5_1, c_1, c9_2


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

(109) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IFPROD(false, nil, x1) -> c4(PRODITER(nil, timesIter(x1, head(nil), 0, 0)))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
K tuples:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c4_1, c5_2, c5_1, c_1, c9_2


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

(111) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
K tuples:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c4_1, c5_2, c5_1, c_1, c9_2


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

(113) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
We considered the (Usable) Rules:
   tail(cons(z0, z1)) -> z1
   tail(nil) -> nil
And the Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IFPROD(x_1, x_2, x_3)) = x_1 + x_2
   POL(IFTIMES(x_1, x_2, x_3, x_4, x_5)) = x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(PRODITER(x_1, x_2)) = x_1
   POL(TIMES(x_1, x_2)) = 0
   POL(TIMESITER(x_1, x_2, x_3, x_4)) = x_4
   POL(c(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(error) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = x_2
   POL(head(x_1)) = 0
   POL(ifTimes(x_1, x_2, x_3, x_4, x_5)) = [1] + x_2 + x_4 + x_5
   POL(nil) = 0
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(times(x_1, x_2)) = [1] + x_1
   POL(timesIter(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_3 + x_4
   POL(true) = [1]

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
K tuples:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c4_1, c5_2, c5_1, c_1, c9_2


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

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error))) by 
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   IFPROD(false, nil, x0) -> c4(PRODITER(nil, times(x0, error)))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, nil, x0) -> c4(PRODITER(nil, times(x0, error)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
K tuples:
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c4_1, c5_2, c5_1, c_1, c9_2


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

(117) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IFPROD(false, nil, x0) -> c4(PRODITER(nil, times(x0, error)))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c4_1, c5_2, c5_1, c_1, c9_2


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

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), times(x1, z0))) by 
   IFPROD(false, cons(z1, x1), z0) -> c4(PRODITER(tail(cons(z1, x1)), timesIter(z0, z1, 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c4_1, c5_2, c5_1, c_1, c9_2


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

(121) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
We considered the (Usable) Rules:
   tail(cons(z0, z1)) -> z1
   tail(nil) -> nil
And the Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IFPROD(x_1, x_2, x_3)) = x_1 + x_2
   POL(IFTIMES(x_1, x_2, x_3, x_4, x_5)) = x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(PRODITER(x_1, x_2)) = x_1
   POL(TIMES(x_1, x_2)) = 0
   POL(TIMESITER(x_1, x_2, x_3, x_4)) = x_4
   POL(c(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(error) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = x_2
   POL(head(x_1)) = 0
   POL(ifTimes(x_1, x_2, x_3, x_4, x_5)) = [1] + x_2 + x_4 + x_5
   POL(nil) = 0
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(times(x_1, x_2)) = [1] + x_1
   POL(timesIter(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_3 + x_4
   POL(true) = [1]

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c4_1, c5_2, c5_1, c_1, c9_2


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

(123) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1))))) by 
   IFPROD(false, cons(x0, x1), z0) -> c4(PRODITER(x1, timesIter(z0, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c5_2, c5_1, c_1, c9_2, c4_1


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

(125) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), timesIter(z0, head(x0), 0, 0)), TIMES(z0, head(x0))) by 
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, nil, x1) -> c5(PRODITER(tail(nil), timesIter(x1, error, 0, 0)), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(PRODITER(nil, timesIter(x1, head(nil), 0, 0)), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, nil, x1) -> c5(PRODITER(tail(nil), timesIter(x1, error, 0, 0)), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(PRODITER(nil, timesIter(x1, head(nil), 0, 0)), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, nil, x1) -> c5(PRODITER(tail(nil), timesIter(x1, error, 0, 0)), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(PRODITER(nil, timesIter(x1, head(nil), 0, 0)), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c5_2, c5_1, c_1, c9_2, c4_1


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

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, nil, x1) -> c5(PRODITER(tail(nil), timesIter(x1, error, 0, 0)), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, nil, x1) -> c5(PRODITER(tail(nil), timesIter(x1, error, 0, 0)), TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c5_2, c5_1, c_1, c9_2, c4_1


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

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c5_2, c5_1, c_1, c9_2, c4_1, c1_1


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

(131) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c5_2, c5_1, c_1, c9_2, c4_1, c1_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.
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
We considered the (Usable) Rules:none
And the Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IFPROD(x_1, x_2, x_3)) = [1] + x_1
   POL(IFTIMES(x_1, x_2, x_3, x_4, x_5)) = x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(PRODITER(x_1, x_2)) = [1]
   POL(TIMES(x_1, x_2)) = 0
   POL(TIMESITER(x_1, x_2, x_3, x_4)) = x_4
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = x_2
   POL(error) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = x_2
   POL(head(x_1)) = 0
   POL(ifTimes(x_1, x_2, x_3, x_4, x_5)) = [1] + x_2 + x_4 + x_5
   POL(nil) = 0
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = 0
   POL(tail(x_1)) = [1] + x_1
   POL(times(x_1, x_2)) = [1] + x_1
   POL(timesIter(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_4
   POL(true) = [1]

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c5_2, c5_1, c_1, c9_2, c4_1, c1_1


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

(135) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
We considered the (Usable) Rules:
   tail(cons(z0, z1)) -> z1
   tail(nil) -> nil
And the Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IFPROD(x_1, x_2, x_3)) = [3]x_1 + [2]x_2
   POL(IFTIMES(x_1, x_2, x_3, x_4, x_5)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(PRODITER(x_1, x_2)) = [2]x_1
   POL(TIMES(x_1, x_2)) = 0
   POL(TIMESITER(x_1, x_2, x_3, x_4)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(error) = 0
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(head(x_1)) = 0
   POL(ifTimes(x_1, x_2, x_3, x_4, x_5)) = [3] + [2]x_2 + [3]x_4 + [3]x_5
   POL(nil) = 0
   POL(plus(x_1, x_2)) = [2] + [3]x_2
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(times(x_1, x_2)) = [3] + [2]x_1
   POL(timesIter(x_1, x_2, x_3, x_4)) = [3]x_1
   POL(true) = [3]

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c5_2, c5_1, c_1, c9_2, c4_1, c1_1


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

(137) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), times(x1, z0)), TIMES(x1, head(cons(z0, z1)))) by 
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), timesIter(z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c5_2, c5_1, c_1, c9_2, c4_1, c1_1


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

(139) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
We considered the (Usable) Rules:
   tail(cons(z0, z1)) -> z1
   tail(nil) -> nil
And the Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IFPROD(x_1, x_2, x_3)) = x_1 + x_2
   POL(IFTIMES(x_1, x_2, x_3, x_4, x_5)) = x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(PRODITER(x_1, x_2)) = x_1
   POL(TIMES(x_1, x_2)) = 0
   POL(TIMESITER(x_1, x_2, x_3, x_4)) = x_4
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(error) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = x_2
   POL(head(x_1)) = 0
   POL(ifTimes(x_1, x_2, x_3, x_4, x_5)) = [1] + x_2 + x_4 + x_5
   POL(nil) = 0
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(times(x_1, x_2)) = [1] + x_1
   POL(timesIter(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_4
   POL(true) = [1]

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c5_2, c5_1, c_1, c9_2, c4_1, c1_1


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

(141) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1)))) by 
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, timesIter(z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c5_1, c_1, c9_2, c4_1, c5_2, c1_1


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

(143) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil))) by 
   IFPROD(false, nil, x0) -> c5(TIMES(x0, error))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, x0) -> c5(TIMES(x0, error))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, times(x1, head(cons(z0, z1)))))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, times(x1, head(cons(z0, z1)))), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c5_1, c_1, c9_2, c4_1, c5_2, c1_1


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

(145) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IFPROD(false, nil, x0) -> c5(TIMES(x0, error))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c_1, c9_2, c4_1, c5_2, c5_1, c1_1


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

(147) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error))) by 
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   IFPROD(false, nil, x0) -> c(PRODITER(nil, times(x0, error)))

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   IFPROD(false, nil, x0) -> c(PRODITER(nil, times(x0, error)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(PRODITER(tail(nil), times(x1, error)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c_1, c9_2, c4_1, c5_2, c5_1, c1_1


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

(149) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IFPROD(false, nil, x0) -> c(PRODITER(nil, times(x0, error)))

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c_1, c9_2, c4_1, c5_2, c5_1, c1_1


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

(151) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil))) by 
   IFPROD(false, nil, x0) -> c(TIMES(x0, error))

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   IFPROD(false, nil, x0) -> c(TIMES(x0, error))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c9_2, c4_1, c5_2, c5_1, c1_1, c_1


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

(153) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IFPROD(false, nil, x0) -> c(TIMES(x0, error))

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c9_2, c4_1, c5_2, c5_1, c1_1, c_1


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

(155) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TIMESITER(s(s(z1)), x1, x2, s(s(z0))) -> c9(IFTIMES(ge(z0, z1), s(s(z1)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(z1)))) by 
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(IFTIMES(true, s(s(0)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(IFTIMES(true, s(s(0)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(IFTIMES(true, s(s(0)), x1, x2, s(s(z0))), GE(s(s(z0)), s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c9_2, c4_1, c5_2, c5_1, c1_1, c_1


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

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c9_2, c4_1, c5_2, c5_1, c1_1, c_1


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

(159) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0))) by IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
----------------------------------------

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c9_2, c4_1, c5_2, c5_1, c1_1, c_1


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

(161) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0))) by IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
----------------------------------------

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c9_2, c4_1, c5_2, c5_1, c1_1, c_1


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

(163) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0))) by IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
----------------------------------------

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c9_2, c4_1, c5_2, c5_1, c1_1, c_1


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

(165) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), timesIter(z0, error, 0, 0))) by IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
----------------------------------------

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c9_2, c4_1, c5_2, c5_1, c1_1, c_1


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

(167) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFPROD(false, cons(z1, x1), z0) -> c4(PRODITER(tail(cons(z1, x1)), timesIter(z0, z1, 0, 0))) by IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
----------------------------------------

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c9_2, c4_1, c5_2, c5_1, c1_1, c_1


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

(169) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0))) by IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
----------------------------------------

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IFTIMES_5, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3

Compound Symbols:   c7_1, c8_1, c11_2, c20_1, c9_1, c2_1, c9_2, c4_1, c5_2, c5_1, c1_1, c_1


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

(171) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IFTIMES(false, z0, z1, z2, z3) -> c11(TIMESITER(z0, z1, plus(z1, z2), s(z3)), PLUS(z1, z2)) by 
   IFTIMES(false, s(x0), x1, x2, 0) -> c11(TIMESITER(s(x0), x1, plus(x1, x2), s(0)), PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c11(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))), PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))), PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c11(TIMESITER(s(x0), x1, plus(x1, x2), s(0)), PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c11(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))), PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))), PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c9(IFTIMES(false, s(s(z0)), x1, x2, s(0)), GE(s(0), s(s(z0))))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c9(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c11(TIMESITER(s(x0), x1, plus(x1, x2), s(0)), PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c11(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))), PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))), PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3, IFTIMES_5

Compound Symbols:   c7_1, c8_1, c20_1, c9_1, c2_1, c9_2, c4_1, c5_2, c5_1, c1_1, c_1, c11_2


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

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, GE_2, TIMESITER_4, PRODITER_2, IFPROD_3, IFTIMES_5

Compound Symbols:   c7_1, c8_1, c20_1, c9_1, c2_1, c4_1, c5_2, c5_1, c1_1, c_1, c9_2, c11_2, c3_1


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

(175) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace TIMESITER(s(z0), x1, x2, 0) -> c9(IFTIMES(false, s(z0), x1, x2, 0)) by 
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))

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

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, GE_2, PRODITER_2, TIMESITER_4, IFPROD_3, IFTIMES_5

Compound Symbols:   c7_1, c8_1, c20_1, c2_1, c9_1, c4_1, c5_2, c5_1, c1_1, c_1, c9_2, c11_2, c3_1


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

(177) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFPROD(false, cons(x0, x1), z0) -> c4(PRODITER(x1, timesIter(z0, head(cons(x0, x1)), 0, 0))) by IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
----------------------------------------

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, GE_2, PRODITER_2, TIMESITER_4, IFPROD_3, IFTIMES_5

Compound Symbols:   c7_1, c8_1, c20_1, c2_1, c9_1, c4_1, c5_2, c5_1, c1_1, c_1, c9_2, c11_2, c3_1


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

(179) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0))) by IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
----------------------------------------

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0)))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, GE_2, PRODITER_2, TIMESITER_4, IFPROD_3, IFTIMES_5

Compound Symbols:   c7_1, c8_1, c20_1, c2_1, c9_1, c4_1, c5_2, c5_1, c1_1, c_1, c9_2, c11_2, c3_1


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

(181) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace TIMESITER(s(x0), x1, x2, s(x3)) -> c9(GE(s(x3), s(x0))) by 
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))

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

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0)))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, GE_2, PRODITER_2, TIMESITER_4, IFPROD_3, IFTIMES_5

Compound Symbols:   c7_1, c8_1, c20_1, c2_1, c9_1, c4_1, c5_2, c5_1, c1_1, c_1, c9_2, c11_2, c3_1


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

(183) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace TIMESITER(s(0), x1, x2, s(z0)) -> c9(GE(s(z0), s(0))) by 
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))

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

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5

Compound Symbols:   c7_1, c8_1, c20_1, c2_1, c4_1, c5_2, c5_1, c1_1, c_1, c9_2, c9_1, c11_2, c3_1


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

(185) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IFPROD(false, x0, z0) -> c4(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0))) by 
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))

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

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5

Compound Symbols:   c7_1, c8_1, c20_1, c2_1, c5_2, c5_1, c1_1, c_1, c9_2, c9_1, c4_1, c11_2, c3_1


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

(187) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IFPROD(false, x0, z0) -> c5(PRODITER(tail(x0), ifTimes(ge(0, z0), z0, head(x0), 0, 0)), TIMES(z0, head(x0))) by 
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))

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

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
S tuples:
   PLUS(s(z0), z1) -> c7(PLUS(z0, z1))
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5

Compound Symbols:   c7_1, c8_1, c20_1, c2_1, c5_2, c5_1, c1_1, c_1, c9_2, c9_1, c4_1, c11_2, c3_1


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

(189) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(z0), z1) -> c7(PLUS(z0, z1)) by 
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))

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

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c5_2, c5_1, c1_1, c_1, c9_2, c9_1, c4_1, c11_2, c3_1, c7_1


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

(191) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1)))) by IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
----------------------------------------

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c5_2, c5_1, c1_1, c_1, c9_2, c9_1, c4_1, c11_2, c3_1, c7_1


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

(193) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1)))) by IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
----------------------------------------

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c5_2, c5_1, c1_1, c_1, c9_2, c9_1, c4_1, c11_2, c3_1, c7_1


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

(195) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil))) by IFPROD(false, nil, z0) -> c5(TIMES(z0, error))
----------------------------------------

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(x1, z0, 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c5(TIMES(z0, error))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, nil, x1) -> c5(TIMES(x1, head(nil)))
   IFPROD(false, nil, x1) -> c4(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, cons(z0, z1), x1) -> c4(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, times(x2, z0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c5_2, c5_1, c1_1, c_1, c9_2, c9_1, c4_1, c11_2, c3_1, c7_1


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

(197) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IFPROD(false, nil, z0) -> c5(TIMES(z0, error))

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

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), timesIter(z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, timesIter(z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c5_1, c1_1, c5_2, c_1, c9_2, c9_1, c4_1, c11_2, c3_1, c7_1


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

(199) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IFPROD(false, x0, x1) -> c5(TIMES(x1, head(x0))) by 
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))

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

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), timesIter(z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, timesIter(z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c1_1, c5_2, c5_1, c_1, c9_2, c9_1, c4_1, c11_2, c3_1, c7_1


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

(201) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0))) by IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
----------------------------------------

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), timesIter(z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, timesIter(z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c1_1, c5_2, c5_1, c_1, c9_2, c9_1, c4_1, c11_2, c3_1, c7_1


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

(203) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil))) by IFPROD(false, nil, z0) -> c1(TIMES(z0, error))
----------------------------------------

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), timesIter(z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, timesIter(z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, nil, z0) -> c1(TIMES(z0, error))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, nil, x1) -> c1(PRODITER(tail(nil), timesIter(x1, error, 0, 0)))
   IFPROD(false, nil, x1) -> c1(TIMES(x1, head(nil)))
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c5_2, c5_1, c_1, c9_2, c9_1, c4_1, c11_2, c3_1, c7_1, c1_1


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

(205) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IFPROD(false, nil, z0) -> c1(TIMES(z0, error))

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

(206)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), timesIter(z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, timesIter(z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   plus_2, tail_1, times_2, head_1, timesIter_4, ifTimes_5, ge_2

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c5_2, c5_1, c_1, c9_2, c9_1, c4_1, c11_2, c3_1, c7_1, c1_1


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

(207) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   head(nil) -> error

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

(208)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   tail(cons(z0, z1)) -> z1
   tail(nil) -> nil
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   head(cons(z0, z1)) -> z0
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), timesIter(z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, timesIter(z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   tail_1, timesIter_4, ifTimes_5, ge_2, plus_2, head_1, times_2

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c5_2, c5_1, c_1, c9_2, c9_1, c4_1, c11_2, c3_1, c7_1, c1_1


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

(209) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), timesIter(z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1)))) by 
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))

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

(210)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   tail(cons(z0, z1)) -> z1
   tail(nil) -> nil
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   head(cons(z0, z1)) -> z0
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, timesIter(z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   tail_1, timesIter_4, ifTimes_5, ge_2, plus_2, head_1, times_2

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c5_2, c5_1, c_1, c9_2, c9_1, c4_1, c11_2, c3_1, c7_1, c1_1


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

(211) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1)))) by 
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(x1, timesIter(z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))

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

(212)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   tail(cons(z0, z1)) -> z1
   tail(nil) -> nil
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   head(cons(z0, z1)) -> z0
   times(z0, z1) -> timesIter(z0, z1, 0, 0)
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, timesIter(z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   tail_1, timesIter_4, ifTimes_5, ge_2, plus_2, head_1, times_2

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c5_1, c5_2, c_1, c9_2, c9_1, c4_1, c11_2, c3_1, c7_1, c1_1


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

(213) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   times(z0, z1) -> timesIter(z0, z1, 0, 0)

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

(214)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, timesIter(z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   head_1, timesIter_4, ifTimes_5, ge_2, plus_2, tail_1

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c5_1, c5_2, c_1, c9_2, c9_1, c4_1, c11_2, c3_1, c7_1, c1_1


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

(215) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1)))) by 
   IFPROD(false, cons(z0, z1), x2) -> c5(TIMES(x2, z0))

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

(216)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, timesIter(z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(TIMES(x2, z0))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   head_1, timesIter_4, ifTimes_5, ge_2, plus_2, tail_1

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c5_2, c_1, c9_2, c9_1, c4_1, c11_2, c3_1, c7_1, c1_1, c5_1


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

(217) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, timesIter(z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1)))) by 
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, ifTimes(ge(0, z0), z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))

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

(218)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0)))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(TIMES(x2, z0))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, ifTimes(ge(0, z0), z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   head_1, timesIter_4, ifTimes_5, ge_2, plus_2, tail_1

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, IFPROD_3, TIMESITER_4, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c_1, c9_2, c9_1, c4_1, c11_2, c3_1, c5_2, c7_1, c1_1, c5_1


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

(219) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), timesIter(z0, error, 0, 0))) by 
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, nil, x0) -> c(PRODITER(nil, timesIter(x0, error, 0, 0)))

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

(220)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(TIMES(x2, z0))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, ifTimes(ge(0, z0), z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, nil, x0) -> c(PRODITER(nil, timesIter(x0, error, 0, 0)))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, cons(z0, z1), x1) -> c5(PRODITER(z1, timesIter(x1, head(cons(z0, z1)), 0, 0)), TIMES(x1, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, times(x2, z0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   head_1, timesIter_4, ifTimes_5, ge_2, plus_2, tail_1

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, TIMESITER_4, IFPROD_3, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c9_2, c9_1, c4_1, c11_2, c3_1, c5_2, c7_1, c1_1, c5_1, c_1


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

(221) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IFPROD(false, nil, x0) -> c(PRODITER(nil, timesIter(x0, error, 0, 0)))

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

(222)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(TIMES(x2, z0))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, ifTimes(ge(0, z0), z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
K tuples:
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   head_1, timesIter_4, ifTimes_5, ge_2, plus_2, tail_1

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, TIMESITER_4, IFPROD_3, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c9_2, c9_1, c4_1, c11_2, c3_1, c5_2, c7_1, c1_1, c5_1, c_1


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

(223) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TIMESITER(s(s(s(z1))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(ge(z0, z1), s(s(s(z1))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1))))) by 
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(true, s(s(s(0))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c9(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))

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

(224)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(TIMES(x2, z0))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, ifTimes(ge(0, z0), z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(true, s(s(s(0))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c9(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(IFTIMES(true, s(s(s(0))), x1, x2, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c9(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
K tuples:
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   head_1, timesIter_4, ifTimes_5, ge_2, plus_2, tail_1

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, TIMESITER_4, IFPROD_3, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c9_1, c4_1, c11_2, c3_1, c5_2, c7_1, c1_1, c5_1, c_1, c9_2


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

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

(226)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(TIMES(x2, z0))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, ifTimes(ge(0, z0), z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c9(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(GE(s(s(s(z0))), s(s(s(0)))))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c9(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(GE(s(s(s(z0))), s(s(s(0)))))
K tuples:
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   head_1, timesIter_4, ifTimes_5, ge_2, plus_2, tail_1

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, TIMESITER_4, IFPROD_3, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c9_1, c4_1, c11_2, c3_1, c5_2, c7_1, c1_1, c5_1, c_1, c9_2


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

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

(228)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(TIMES(x2, z0))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, ifTimes(ge(0, z0), z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(GE(s(s(s(0))), s(s(s(s(z0))))))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(GE(s(s(s(0))), s(s(s(s(z0))))))
K tuples:
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   head_1, timesIter_4, ifTimes_5, ge_2, plus_2, tail_1

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, TIMESITER_4, IFPROD_3, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c9_1, c4_1, c11_2, c3_1, c5_2, c7_1, c1_1, c5_1, c_1, c9_2, c6_1


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

(229) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, nil, z0) -> c4(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0))) by 
   IFPROD(false, nil, 0) -> c4(PRODITER(tail(nil), ifTimes(true, 0, error, 0, 0)))
   IFPROD(false, nil, s(z0)) -> c4(PRODITER(tail(nil), ifTimes(false, s(z0), error, 0, 0)))
   IFPROD(false, nil, x0) -> c4(PRODITER(nil, ifTimes(ge(0, x0), x0, error, 0, 0)))

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

(230)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(TIMES(x2, z0))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, ifTimes(ge(0, z0), z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(GE(s(s(s(0))), s(s(s(s(z0))))))
   IFPROD(false, nil, 0) -> c4(PRODITER(tail(nil), ifTimes(true, 0, error, 0, 0)))
   IFPROD(false, nil, s(z0)) -> c4(PRODITER(tail(nil), ifTimes(false, s(z0), error, 0, 0)))
   IFPROD(false, nil, x0) -> c4(PRODITER(nil, ifTimes(ge(0, x0), x0, error, 0, 0)))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(GE(s(s(s(0))), s(s(s(s(z0))))))
K tuples:
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   head_1, timesIter_4, ifTimes_5, ge_2, plus_2, tail_1

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, TIMESITER_4, IFPROD_3, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c9_1, c4_1, c11_2, c3_1, c5_2, c7_1, c1_1, c5_1, c_1, c9_2, c6_1


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

(231) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IFPROD(false, nil, x0) -> c4(PRODITER(nil, ifTimes(ge(0, x0), x0, error, 0, 0)))

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

(232)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(TIMES(x2, z0))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, ifTimes(ge(0, z0), z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(GE(s(s(s(0))), s(s(s(s(z0))))))
   IFPROD(false, nil, 0) -> c4(PRODITER(tail(nil), ifTimes(true, 0, error, 0, 0)))
   IFPROD(false, nil, s(z0)) -> c4(PRODITER(tail(nil), ifTimes(false, s(z0), error, 0, 0)))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(GE(s(s(s(0))), s(s(s(s(z0))))))
K tuples:
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   head_1, timesIter_4, ifTimes_5, ge_2, plus_2, tail_1

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, TIMESITER_4, IFPROD_3, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c9_1, c4_1, c11_2, c3_1, c5_2, c7_1, c1_1, c5_1, c_1, c9_2, c6_1


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

(233) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(tail(cons(z0, z1)), ifTimes(ge(0, z2), z2, z0, 0, 0))) by 
   IFPROD(false, cons(x0, x1), 0) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(true, 0, x0, 0, 0)))
   IFPROD(false, cons(x0, x1), s(z0)) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(false, s(z0), x0, 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, ifTimes(ge(0, x2), x2, z0, 0, 0)))

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

(234)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(TIMES(x2, z0))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, ifTimes(ge(0, z0), z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(GE(s(s(s(0))), s(s(s(s(z0))))))
   IFPROD(false, nil, 0) -> c4(PRODITER(tail(nil), ifTimes(true, 0, error, 0, 0)))
   IFPROD(false, nil, s(z0)) -> c4(PRODITER(tail(nil), ifTimes(false, s(z0), error, 0, 0)))
   IFPROD(false, cons(x0, x1), 0) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(true, 0, x0, 0, 0)))
   IFPROD(false, cons(x0, x1), s(z0)) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(false, s(z0), x0, 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, ifTimes(ge(0, x2), x2, z0, 0, 0)))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(GE(s(s(s(0))), s(s(s(s(z0))))))
   IFPROD(false, cons(x0, x1), 0) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(true, 0, x0, 0, 0)))
   IFPROD(false, cons(x0, x1), s(z0)) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(false, s(z0), x0, 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, ifTimes(ge(0, x2), x2, z0, 0, 0)))
K tuples:
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
Defined Rule Symbols:   head_1, timesIter_4, ifTimes_5, ge_2, plus_2, tail_1

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, TIMESITER_4, IFPROD_3, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c9_1, c4_1, c11_2, c3_1, c5_2, c7_1, c1_1, c5_1, c_1, c9_2, c6_1


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

(235) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, ifTimes(ge(0, x2), x2, z0, 0, 0)))
We considered the (Usable) Rules:
   tail(cons(z0, z1)) -> z1
   tail(nil) -> nil
And the Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(TIMES(x2, z0))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, ifTimes(ge(0, z0), z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(GE(s(s(s(0))), s(s(s(s(z0))))))
   IFPROD(false, nil, 0) -> c4(PRODITER(tail(nil), ifTimes(true, 0, error, 0, 0)))
   IFPROD(false, nil, s(z0)) -> c4(PRODITER(tail(nil), ifTimes(false, s(z0), error, 0, 0)))
   IFPROD(false, cons(x0, x1), 0) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(true, 0, x0, 0, 0)))
   IFPROD(false, cons(x0, x1), s(z0)) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(false, s(z0), x0, 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, ifTimes(ge(0, x2), x2, z0, 0, 0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IFPROD(x_1, x_2, x_3)) = x_1 + x_2
   POL(IFTIMES(x_1, x_2, x_3, x_4, x_5)) = x_2 + x_5
   POL(PLUS(x_1, x_2)) = 0
   POL(PRODITER(x_1, x_2)) = x_1
   POL(TIMES(x_1, x_2)) = 0
   POL(TIMESITER(x_1, x_2, x_3, x_4)) = x_4
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(error) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(head(x_1)) = [1]
   POL(ifTimes(x_1, x_2, x_3, x_4, x_5)) = [1] + x_2 + x_4 + x_5
   POL(nil) = 0
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(timesIter(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_3
   POL(true) = [1]

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

(236)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0)))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(TIMES(x2, z0))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, ifTimes(ge(0, z0), z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(GE(s(s(s(0))), s(s(s(s(z0))))))
   IFPROD(false, nil, 0) -> c4(PRODITER(tail(nil), ifTimes(true, 0, error, 0, 0)))
   IFPROD(false, nil, s(z0)) -> c4(PRODITER(tail(nil), ifTimes(false, s(z0), error, 0, 0)))
   IFPROD(false, cons(x0, x1), 0) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(true, 0, x0, 0, 0)))
   IFPROD(false, cons(x0, x1), s(z0)) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(false, s(z0), x0, 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, ifTimes(ge(0, x2), x2, z0, 0, 0)))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(GE(s(s(s(0))), s(s(s(s(z0))))))
   IFPROD(false, cons(x0, x1), 0) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(true, 0, x0, 0, 0)))
   IFPROD(false, cons(x0, x1), s(z0)) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(false, s(z0), x0, 0, 0)))
K tuples:
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, ifTimes(ge(0, x2), x2, z0, 0, 0)))
Defined Rule Symbols:   head_1, timesIter_4, ifTimes_5, ge_2, plus_2, tail_1

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, TIMESITER_4, IFPROD_3, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c9_1, c4_1, c11_2, c3_1, c5_2, c7_1, c1_1, c5_1, c_1, c9_2, c6_1


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

(237) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, ifTimes(ge(0, z2), z2, head(cons(z0, z1)), 0, 0))) by 
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, ifTimes(ge(0, x2), x2, z0, 0, 0)))
   IFPROD(false, cons(x0, x1), 0) -> c4(PRODITER(x1, ifTimes(true, 0, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), s(z0)) -> c4(PRODITER(x1, ifTimes(false, s(z0), head(cons(x0, x1)), 0, 0)))

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

(238)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   head(cons(z0, z1)) -> z0
   timesIter(z0, z1, z2, z3) -> ifTimes(ge(z3, z0), z0, z1, z2, z3)
   ifTimes(true, z0, z1, z2, z3) -> z2
   ifTimes(false, z0, z1, z2, z3) -> timesIter(z0, z1, plus(z1, z2), s(z3))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFPROD(false, cons(z0, z1), z2) -> c4(PRODITER(z1, timesIter(z2, z0, 0, 0)))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(z1, timesIter(z2, head(cons(z0, z1)), 0, 0)), TIMES(z2, z0))
   IFPROD(false, nil, z0) -> c1(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   IFPROD(false, cons(z1, x1), z0) -> c5(PRODITER(tail(cons(z1, x1)), ifTimes(ge(0, z0), z0, z1, 0, 0)), TIMES(z0, head(cons(z1, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(PRODITER(z1, timesIter(x2, z0, 0, 0)), TIMES(x2, head(cons(z0, z1))))
   IFPROD(false, cons(z0, z1), x2) -> c5(TIMES(x2, z0))
   IFPROD(false, cons(x0, x1), z0) -> c5(PRODITER(x1, ifTimes(ge(0, z0), z0, head(cons(x0, x1)), 0, 0)), TIMES(z0, head(cons(x0, x1))))
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, nil, z0) -> c(PRODITER(tail(nil), ifTimes(ge(0, z0), z0, error, 0, 0)))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(GE(s(s(s(0))), s(s(s(s(z0))))))
   IFPROD(false, nil, 0) -> c4(PRODITER(tail(nil), ifTimes(true, 0, error, 0, 0)))
   IFPROD(false, nil, s(z0)) -> c4(PRODITER(tail(nil), ifTimes(false, s(z0), error, 0, 0)))
   IFPROD(false, cons(x0, x1), 0) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(true, 0, x0, 0, 0)))
   IFPROD(false, cons(x0, x1), s(z0)) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(false, s(z0), x0, 0, 0)))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, ifTimes(ge(0, x2), x2, z0, 0, 0)))
   IFPROD(false, cons(x0, x1), 0) -> c4(PRODITER(x1, ifTimes(true, 0, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), s(z0)) -> c4(PRODITER(x1, ifTimes(false, s(z0), head(cons(x0, x1)), 0, 0)))
S tuples:
   TIMES(z0, z1) -> c8(TIMESITER(z0, z1, 0, 0))
   GE(s(z0), s(z1)) -> c20(GE(z0, z1))
   PRODITER(cons(z0, z1), x1) -> c2(IFPROD(false, cons(z0, z1), x1))
   TIMESITER(s(s(x0)), x1, x2, s(s(x3))) -> c9(GE(s(s(x3)), s(s(x0))))
   TIMESITER(s(s(0)), x1, x2, s(s(z0))) -> c9(GE(s(s(z0)), s(s(0))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c11(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(s(x3))))), PLUS(x1, x2))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(IFTIMES(false, s(s(z0)), x1, x2, s(0)))
   TIMESITER(s(s(z0)), x1, x2, s(0)) -> c3(GE(s(0), s(s(z0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(IFTIMES(false, s(s(s(z0))), x1, x2, s(s(0))))
   TIMESITER(s(s(s(z0))), x1, x2, s(s(0))) -> c3(GE(s(s(0)), s(s(s(z0)))))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(TIMESITER(s(x0), x1, plus(x1, x2), s(0)))
   IFTIMES(false, s(x0), x1, x2, 0) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(TIMESITER(s(s(x0)), x1, plus(x1, x2), s(s(0))))
   IFTIMES(false, s(s(x0)), x1, x2, s(0)) -> c3(PLUS(x1, x2))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(TIMESITER(s(s(s(x0))), x1, plus(x1, x2), s(s(s(0)))))
   IFTIMES(false, s(s(s(x0))), x1, x2, s(s(0))) -> c3(PLUS(x1, x2))
   TIMESITER(s(z0), x1, 0, 0) -> c9(IFTIMES(false, s(z0), x1, 0, 0))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(s(x3))))) -> c9(GE(s(s(s(s(x3)))), s(s(s(x0)))))
   TIMESITER(s(x0), x1, y0, s(0)) -> c9(GE(s(0), s(x0)))
   TIMESITER(s(s(x0)), x1, y0, s(s(0))) -> c9(GE(s(s(0)), s(s(x0))))
   TIMESITER(s(s(s(x0))), x1, y0, s(s(s(0)))) -> c9(GE(s(s(s(0))), s(s(s(x0)))))
   TIMESITER(s(0), x1, y0, s(0)) -> c9(GE(s(0), s(0)))
   IFPROD(false, cons(x0, x1), x2) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)))
   IFPROD(false, cons(x0, x1), x2) -> c5(PRODITER(tail(cons(x0, x1)), ifTimes(ge(0, x2), x2, head(cons(x0, x1)), 0, 0)), TIMES(x2, head(cons(x0, x1))))
   PLUS(s(s(y0)), z1) -> c7(PLUS(s(y0), z1))
   IFPROD(false, cons(z0, z1), z2) -> c5(PRODITER(tail(cons(z0, z1)), timesIter(z2, z0, 0, 0)), TIMES(z2, z0))
   TIMESITER(s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))) -> c9(IFTIMES(ge(z0, z1), s(s(s(s(z1)))), x1, x2, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   TIMESITER(s(s(s(x0))), x1, x2, s(s(s(x3)))) -> c9(GE(s(s(s(x3))), s(s(s(x0)))))
   TIMESITER(s(s(s(0))), x1, x2, s(s(s(z0)))) -> c9(GE(s(s(s(z0))), s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(IFTIMES(false, s(s(s(s(z0)))), x1, x2, s(s(s(0)))))
   TIMESITER(s(s(s(s(z0)))), x1, x2, s(s(s(0)))) -> c6(GE(s(s(s(0))), s(s(s(s(z0))))))
   IFPROD(false, cons(x0, x1), 0) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(true, 0, x0, 0, 0)))
   IFPROD(false, cons(x0, x1), s(z0)) -> c4(PRODITER(tail(cons(x0, x1)), ifTimes(false, s(z0), x0, 0, 0)))
K tuples:
   IFPROD(false, cons(x0, x1), x2) -> c5(TIMES(x2, head(cons(x0, x1))))
   IFPROD(false, cons(z0, z1), x2) -> c4(PRODITER(z1, ifTimes(ge(0, x2), x2, z0, 0, 0)))
Defined Rule Symbols:   head_1, timesIter_4, ifTimes_5, ge_2, plus_2, tail_1

Defined Pair Symbols:   TIMES_2, GE_2, PRODITER_2, TIMESITER_4, IFPROD_3, IFTIMES_5, PLUS_2

Compound Symbols:   c8_1, c20_1, c2_1, c9_1, c4_1, c11_2, c3_1, c5_2, c7_1, c1_1, c5_1, c_1, c9_2, c6_1