KILLED
proof of input_hP6Z5hO8jG.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 5 ms]
        (14) CpxRNTS
        (15) InliningProof [UPPER BOUND(ID), 358 ms]
        (16) CpxRNTS
        (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 894 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 356 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 390 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 36 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 322 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 84 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 42 ms]
        (44) CpxRNTS
        (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 117 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
        (50) CpxRNTS
        (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (52) CpxRNTS
        (53) IntTrsBoundProof [UPPER BOUND(ID), 229 ms]
        (54) CpxRNTS
        (55) IntTrsBoundProof [UPPER BOUND(ID), 64 ms]
        (56) CpxRNTS
        (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (58) CpxRNTS
        (59) IntTrsBoundProof [UPPER BOUND(ID), 14.3 s]
        (60) CpxRNTS
        (61) IntTrsBoundProof [UPPER BOUND(ID), 2736 ms]
        (62) CpxRNTS
    (63) CompletionProof [UPPER BOUND(ID), 0 ms]
    (64) CpxTypedWeightedCompleteTrs
        (65) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 5 ms]
        (66) CpxRNTS
(67) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(68) CdtProblem
    (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (76) CdtProblem
    (77) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 7 ms]
    (80) CdtProblem
    (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 55 ms]
    (90) CdtProblem
    (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 42 ms]
    (98) CdtProblem
    (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 38 ms]
    (114) CdtProblem
    (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 55 ms]
    (120) CdtProblem
    (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 54 ms]
    (126) CdtProblem
    (127) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 51 ms]
    (128) CdtProblem
    (129) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 59 ms]
    (130) CdtProblem
    (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 43 ms]
    (138) CdtProblem
    (139) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 39 ms]
    (140) CdtProblem
    (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 50 ms]
    (146) CdtProblem
    (147) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 57 ms]
    (148) CdtProblem
    (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 50 ms]
    (154) CdtProblem
    (155) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 57 ms]
    (156) CdtProblem
    (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 67 ms]
    (164) CdtProblem
    (165) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 45 ms]
    (166) CdtProblem
    (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem
    (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (176) CdtProblem
    (177) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (178) CdtProblem
    (179) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (180) CdtProblem
    (181) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (182) CdtProblem
    (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (184) CdtProblem
    (185) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (186) CdtProblem
    (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (188) CdtProblem
    (189) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (190) CdtProblem
    (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (192) CdtProblem
    (193) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (194) CdtProblem
    (195) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (196) CdtProblem
    (197) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (198) CdtProblem
    (199) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (200) CdtProblem
    (201) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (202) CdtProblem
    (203) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (204) CdtProblem
    (205) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 9 ms]
    (206) CdtProblem
    (207) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (208) CdtProblem
    (209) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (210) CdtProblem
    (211) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (212) CdtProblem
    (213) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 23 ms]
    (214) CdtProblem
    (215) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (216) CdtProblem
    (217) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (218) CdtProblem
    (219) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (220) CdtProblem
    (221) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 16 ms]
    (222) CdtProblem
    (223) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (224) CdtProblem
    (225) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (226) CdtProblem
    (227) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (228) CdtProblem
    (229) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (230) CdtProblem
    (231) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (232) CdtProblem
    (233) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (234) CdtProblem
    (235) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (236) CdtProblem
    (237) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (238) CdtProblem
    (239) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (240) CdtProblem
    (241) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (242) CdtProblem
    (243) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 44 ms]
    (244) CdtProblem
    (245) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (246) CdtProblem
    (247) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (248) CdtProblem
    (249) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (250) CdtProblem
    (251) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (252) CdtProblem
    (253) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (254) CdtProblem
    (255) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (256) CdtProblem
    (257) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (258) CdtProblem
    (259) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (260) CdtProblem
    (261) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (262) CdtProblem
    (263) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (264) CdtProblem
    (265) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (266) CdtProblem
    (267) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (268) CdtProblem
    (269) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (270) CdtProblem
    (271) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (272) 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:

   lcm(x, y) -> lcmIter(x, y, 0, times(x, y))
   lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u)
   if(true, x, y, z, u) -> z
   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
   if2(true, x, y, z, u) -> z
   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> ifTimes(ge(0, x), x, y)
   ifTimes(true, x, y) -> 0
   ifTimes(false, x, y) -> plus(y, times(y, p(x)))
   p(s(x)) -> x
   p(0) -> s(s(0))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   or(true, y) -> true
   or(false, y) -> y
   divisible(0, s(y)) -> true
   divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
   div(x, y, 0) -> divisible(x, y)
   div(0, y, s(z)) -> false
   div(s(x), y, s(z)) -> div(x, y, z)
   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:

   lcm(x, y) -> lcmIter(x, y, 0', times(x, y))
   lcmIter(x, y, z, u) -> if(or(ge(0', x), ge(z, u)), x, y, z, u)
   if(true, x, y, z, u) -> z
   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
   if2(true, x, y, z, u) -> z
   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> ifTimes(ge(0', x), x, y)
   ifTimes(true, x, y) -> 0'
   ifTimes(false, x, y) -> plus(y, times(y, p(x)))
   p(s(x)) -> x
   p(0') -> s(s(0'))
   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   or(true, y) -> true
   or(false, y) -> y
   divisible(0', s(y)) -> true
   divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
   div(x, y, 0') -> divisible(x, y)
   div(0', y, s(z)) -> false
   div(s(x), y, s(z)) -> div(x, y, z)
   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:

   lcm(x, y) -> lcmIter(x, y, 0, times(x, y))
   lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u)
   if(true, x, y, z, u) -> z
   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
   if2(true, x, y, z, u) -> z
   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> ifTimes(ge(0, x), x, y)
   ifTimes(true, x, y) -> 0
   ifTimes(false, x, y) -> plus(y, times(y, p(x)))
   p(s(x)) -> x
   p(0) -> s(s(0))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   or(true, y) -> true
   or(false, y) -> y
   divisible(0, s(y)) -> true
   divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
   div(x, y, 0) -> divisible(x, y)
   div(0, y, s(z)) -> false
   div(s(x), y, s(z)) -> div(x, y, z)
   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:

   lcm(x, y) -> lcmIter(x, y, 0, times(x, y)) [1]
   lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u) [1]
   if(true, x, y, z, u) -> z [1]
   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u) [1]
   if2(true, x, y, z, u) -> z [1]
   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(x, y) -> ifTimes(ge(0, x), x, y) [1]
   ifTimes(true, x, y) -> 0 [1]
   ifTimes(false, x, y) -> plus(y, times(y, p(x))) [1]
   p(s(x)) -> x [1]
   p(0) -> s(s(0)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   or(true, y) -> true [1]
   or(false, y) -> y [1]
   divisible(0, s(y)) -> true [1]
   divisible(s(x), s(y)) -> div(s(x), s(y), s(y)) [1]
   div(x, y, 0) -> divisible(x, y) [1]
   div(0, y, s(z)) -> false [1]
   div(s(x), y, s(z)) -> div(x, y, z) [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:

   lcm(x, y) -> lcmIter(x, y, 0, times(x, y)) [1]
   lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u) [1]
   if(true, x, y, z, u) -> z [1]
   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u) [1]
   if2(true, x, y, z, u) -> z [1]
   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(x, y) -> ifTimes(ge(0, x), x, y) [1]
   ifTimes(true, x, y) -> 0 [1]
   ifTimes(false, x, y) -> plus(y, times(y, p(x))) [1]
   p(s(x)) -> x [1]
   p(0) -> s(s(0)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   or(true, y) -> true [1]
   or(false, y) -> y [1]
   divisible(0, s(y)) -> true [1]
   divisible(s(x), s(y)) -> div(s(x), s(y), s(y)) [1]
   div(x, y, 0) -> divisible(x, y) [1]
   div(0, y, s(z)) -> false [1]
   div(s(x), y, s(z)) -> div(x, y, z) [1]
   a -> b [1]
   a -> c [1]

The TRS has the following type information:
lcm :: 0:s -> 0:s -> 0:s
lcmIter :: 0:s -> 0:s -> 0:s -> 0:s -> 0:s
0 :: 0:s
times :: 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s
or :: true:false -> true:false -> true:false
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
if2 :: true:false -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s
divisible :: 0:s -> 0:s -> true:false
plus :: 0:s -> 0:s -> 0:s
s :: 0:s -> 0:s
ifTimes :: true:false -> 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s -> true:false
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:

   lcm_2
   lcmIter_4
   if_5
   if2_5
   a

(c) The following functions are completely defined:

   times_2
   divisible_2
   plus_2
   or_2
   ge_2
   p_1
   div_3
   ifTimes_3

Due to the following rules being added:

   divisible(v0, v1) -> null_divisible [0]
   or(v0, v1) -> null_or [0]
   ifTimes(v0, v1, v2) -> 0 [0]

And the following fresh constants:    null_divisible, null_or

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

(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:

   lcm(x, y) -> lcmIter(x, y, 0, times(x, y)) [1]
   lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u) [1]
   if(true, x, y, z, u) -> z [1]
   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u) [1]
   if2(true, x, y, z, u) -> z [1]
   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(x, y) -> ifTimes(ge(0, x), x, y) [1]
   ifTimes(true, x, y) -> 0 [1]
   ifTimes(false, x, y) -> plus(y, times(y, p(x))) [1]
   p(s(x)) -> x [1]
   p(0) -> s(s(0)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   or(true, y) -> true [1]
   or(false, y) -> y [1]
   divisible(0, s(y)) -> true [1]
   divisible(s(x), s(y)) -> div(s(x), s(y), s(y)) [1]
   div(x, y, 0) -> divisible(x, y) [1]
   div(0, y, s(z)) -> false [1]
   div(s(x), y, s(z)) -> div(x, y, z) [1]
   a -> b [1]
   a -> c [1]
   divisible(v0, v1) -> null_divisible [0]
   or(v0, v1) -> null_or [0]
   ifTimes(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
lcm :: 0:s -> 0:s -> 0:s
lcmIter :: 0:s -> 0:s -> 0:s -> 0:s -> 0:s
0 :: 0:s
times :: 0:s -> 0:s -> 0:s
if :: true:false:null_divisible:null_or -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s
or :: true:false:null_divisible:null_or -> true:false:null_divisible:null_or -> true:false:null_divisible:null_or
ge :: 0:s -> 0:s -> true:false:null_divisible:null_or
true :: true:false:null_divisible:null_or
false :: true:false:null_divisible:null_or
if2 :: true:false:null_divisible:null_or -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s
divisible :: 0:s -> 0:s -> true:false:null_divisible:null_or
plus :: 0:s -> 0:s -> 0:s
s :: 0:s -> 0:s
ifTimes :: true:false:null_divisible:null_or -> 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s -> true:false:null_divisible:null_or
a :: b:c
b :: b:c
c :: b:c
null_divisible :: true:false:null_divisible:null_or
null_or :: true:false:null_divisible:null_or

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:

   lcm(x, y) -> lcmIter(x, y, 0, ifTimes(ge(0, x), x, y)) [2]
   lcmIter(0, y, z, 0) -> if(or(true, true), 0, y, z, 0) [3]
   lcmIter(0, y, 0, s(y'')) -> if(or(true, false), 0, y, 0, s(y'')) [3]
   lcmIter(0, y, s(x'), s(y1)) -> if(or(true, ge(x', y1)), 0, y, s(x'), s(y1)) [3]
   lcmIter(s(y'), y, z, 0) -> if(or(false, true), s(y'), y, z, 0) [3]
   lcmIter(s(y'), y, 0, s(y2)) -> if(or(false, false), s(y'), y, 0, s(y2)) [3]
   lcmIter(s(y'), y, s(x''), s(y3)) -> if(or(false, ge(x'', y3)), s(y'), y, s(x''), s(y3)) [3]
   if(true, x, y, z, u) -> z [1]
   if(false, x, s(y4), 0, u) -> if2(true, x, s(y4), 0, u) [2]
   if(false, x, s(y5), s(x1), u) -> if2(div(s(x1), s(y5), s(y5)), x, s(y5), s(x1), u) [2]
   if(false, x, y, z, u) -> if2(null_divisible, x, y, z, u) [1]
   if2(true, x, y, z, u) -> z [1]
   if2(false, 0, y, z, u) -> lcmIter(0, y, z, u) [2]
   if2(false, s(x2), y, z, u) -> lcmIter(s(x2), y, s(plus(x2, z)), u) [2]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(0, y) -> ifTimes(true, 0, y) [2]
   times(s(y6), y) -> ifTimes(false, s(y6), y) [2]
   ifTimes(true, x, y) -> 0 [1]
   ifTimes(false, s(x3), y) -> plus(y, times(y, x3)) [2]
   ifTimes(false, 0, y) -> plus(y, times(y, s(s(0)))) [2]
   p(s(x)) -> x [1]
   p(0) -> s(s(0)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   or(true, y) -> true [1]
   or(false, y) -> y [1]
   divisible(0, s(y)) -> true [1]
   divisible(s(x), s(y)) -> div(s(x), s(y), s(y)) [1]
   div(x, y, 0) -> divisible(x, y) [1]
   div(0, y, s(z)) -> false [1]
   div(s(x), y, s(z)) -> div(x, y, z) [1]
   a -> b [1]
   a -> c [1]
   divisible(v0, v1) -> null_divisible [0]
   or(v0, v1) -> null_or [0]
   ifTimes(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
lcm :: 0:s -> 0:s -> 0:s
lcmIter :: 0:s -> 0:s -> 0:s -> 0:s -> 0:s
0 :: 0:s
times :: 0:s -> 0:s -> 0:s
if :: true:false:null_divisible:null_or -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s
or :: true:false:null_divisible:null_or -> true:false:null_divisible:null_or -> true:false:null_divisible:null_or
ge :: 0:s -> 0:s -> true:false:null_divisible:null_or
true :: true:false:null_divisible:null_or
false :: true:false:null_divisible:null_or
if2 :: true:false:null_divisible:null_or -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s
divisible :: 0:s -> 0:s -> true:false:null_divisible:null_or
plus :: 0:s -> 0:s -> 0:s
s :: 0:s -> 0:s
ifTimes :: true:false:null_divisible:null_or -> 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s -> true:false:null_divisible:null_or
a :: b:c
b :: b:c
c :: b:c
null_divisible :: true:false:null_divisible:null_or
null_or :: true:false:null_divisible:null_or

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
   b => 0
   c => 1
   null_divisible => 0
   null_or => 0

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 1 }-> divisible(x, y) :|: z1 = 0, z' = x, z'' = y, x >= 0, y >= 0
   div(z', z'', z1) -{ 1 }-> div(x, y, z) :|: z' = 1 + x, z >= 0, z'' = y, x >= 0, y >= 0, z1 = 1 + z
   div(z', z'', z1) -{ 1 }-> 1 :|: z >= 0, z'' = y, y >= 0, z1 = 1 + z, z' = 0
   divisible(z', z'') -{ 1 }-> div(1 + x, 1 + y, 1 + y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   divisible(z', z'') -{ 1 }-> 2 :|: y >= 0, z'' = 1 + y, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   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
   if(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
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(div(1 + x1, 1 + y5, 1 + y5), x, 1 + y5, 1 + x1, u) :|: z1 = 1 + y5, y5 >= 0, x1 >= 0, x >= 0, z'' = x, z2 = 1 + x1, z' = 1, z3 = u, u >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, x, 1 + y4, 0, u) :|: z2 = 0, x >= 0, z1 = 1 + y4, z'' = x, z' = 1, y4 >= 0, z3 = u, u >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, x, y, z, u) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1, z3 = u, u >= 0
   if2(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
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, y, z, u) :|: z'' = 0, z1 = y, z >= 0, z2 = z, y >= 0, z' = 1, z3 = u, u >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(1 + x2, y, 1 + plus(x2, z), u) :|: z1 = y, z >= 0, z2 = z, y >= 0, z'' = 1 + x2, z' = 1, z3 = u, x2 >= 0, u >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(y, times(y, x3)) :|: z1 = y, z'' = 1 + x3, y >= 0, z' = 1, x3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(y, times(y, 1 + (1 + 0))) :|: z'' = 0, z1 = y, y >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z1 = y, z' = 2, x >= 0, y >= 0, z'' = x
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   lcm(z', z'') -{ 2 }-> lcmIter(x, y, 0, ifTimes(ge(0, x), x, y)) :|: z' = x, z'' = y, x >= 0, y >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(2, ge(x', y1)), 0, y, 1 + x', 1 + y1) :|: y1 >= 0, z2 = 1 + y1, z'' = y, y >= 0, x' >= 0, z1 = 1 + x', z' = 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(2, 2), 0, y, z, 0) :|: z1 = z, z >= 0, z'' = y, z2 = 0, y >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(2, 1), 0, y, 0, 1 + y'') :|: z2 = 1 + y'', z1 = 0, z'' = y, y >= 0, y'' >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(1, ge(x'', y3)), 1 + y', y, 1 + x'', 1 + y3) :|: z1 = 1 + x'', z2 = 1 + y3, z'' = y, y >= 0, y' >= 0, y3 >= 0, z' = 1 + y', x'' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(1, 2), 1 + y', y, z, 0) :|: z1 = z, z >= 0, z'' = y, z2 = 0, y >= 0, y' >= 0, z' = 1 + y'
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(1, 1), 1 + y', y, 0, 1 + y2) :|: z1 = 0, z2 = 1 + y2, z'' = y, y >= 0, y' >= 0, z' = 1 + y', y2 >= 0
   or(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' = y, y >= 0
   or(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, y) :|: z'' = y, y >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + y6, y) :|: z'' = y, y >= 0, y6 >= 0, z' = 1 + y6


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

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

   or(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 1
   or(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' = y, y >= 0

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 1 }-> divisible(x, y) :|: z1 = 0, z' = x, z'' = y, x >= 0, y >= 0
   div(z', z'', z1) -{ 1 }-> div(x, y, z) :|: z' = 1 + x, z >= 0, z'' = y, x >= 0, y >= 0, z1 = 1 + z
   div(z', z'', z1) -{ 1 }-> 1 :|: z >= 0, z'' = y, y >= 0, z1 = 1 + z, z' = 0
   divisible(z', z'') -{ 1 }-> div(1 + x, 1 + y, 1 + y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   divisible(z', z'') -{ 1 }-> 2 :|: y >= 0, z'' = 1 + y, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   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
   if(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
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(div(1 + x1, 1 + y5, 1 + y5), x, 1 + y5, 1 + x1, u) :|: z1 = 1 + y5, y5 >= 0, x1 >= 0, x >= 0, z'' = x, z2 = 1 + x1, z' = 1, z3 = u, u >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, x, 1 + y4, 0, u) :|: z2 = 0, x >= 0, z1 = 1 + y4, z'' = x, z' = 1, y4 >= 0, z3 = u, u >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, x, y, z, u) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1, z3 = u, u >= 0
   if2(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
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, y, z, u) :|: z'' = 0, z1 = y, z >= 0, z2 = z, y >= 0, z' = 1, z3 = u, u >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(1 + x2, y, 1 + plus(x2, z), u) :|: z1 = y, z >= 0, z2 = z, y >= 0, z'' = 1 + x2, z' = 1, z3 = u, x2 >= 0, u >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(y, times(y, x3)) :|: z1 = y, z'' = 1 + x3, y >= 0, z' = 1, x3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(y, times(y, 1 + (1 + 0))) :|: z'' = 0, z1 = y, y >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z1 = y, z' = 2, x >= 0, y >= 0, z'' = x
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   lcm(z', z'') -{ 2 }-> lcmIter(x, y, 0, ifTimes(ge(0, x), x, y)) :|: z' = x, z'' = y, x >= 0, y >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + y', y, z, 0) :|: z1 = z, z >= 0, z'' = y, z2 = 0, y >= 0, y' >= 0, z' = 1 + y', 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + y', y, 0, 1 + y2) :|: z1 = 0, z2 = 1 + y2, z'' = y, y >= 0, y' >= 0, z' = 1 + y', y2 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(2, ge(x', y1)), 0, y, 1 + x', 1 + y1) :|: y1 >= 0, z2 = 1 + y1, z'' = y, y >= 0, x' >= 0, z1 = 1 + x', z' = 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(1, ge(x'', y3)), 1 + y', y, 1 + x'', 1 + y3) :|: z1 = 1 + x'', z2 = 1 + y3, z'' = y, y >= 0, y' >= 0, y3 >= 0, z' = 1 + y', x'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, y, z, 0) :|: z1 = z, z >= 0, z'' = y, z2 = 0, y >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, y, 0, 1 + y'') :|: z2 = 1 + y'', z1 = 0, z'' = y, y >= 0, y'' >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, y, z, 0) :|: z1 = z, z >= 0, z'' = y, z2 = 0, y >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, y, 0, 1 + y'') :|: z2 = 1 + y'', z1 = 0, z'' = y, y >= 0, y'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + y', y, z, 0) :|: z1 = z, z >= 0, z'' = y, z2 = 0, y >= 0, y' >= 0, z' = 1 + y', v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + y', y, 0, 1 + y2) :|: z1 = 0, z2 = 1 + y2, z'' = y, y >= 0, y' >= 0, z' = 1 + y', y2 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' = y, y >= 0
   or(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, y) :|: z'' = y, y >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + y6, y) :|: z'' = y, y >= 0, y6 >= 0, z' = 1 + y6


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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 1 }-> divisible(z', z'') :|: z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> div(z' - 1, z'', z1 - 1) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 1 }-> div(1 + (z' - 1), 1 + (z'' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(div(1 + (z2 - 1), 1 + (z1 - 1), 1 + (z1 - 1)), z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(1 + (z'' - 1), z1, 1 + plus(z'' - 1, z2), z3) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 2 }-> lcmIter(z', z'', 0, ifTimes(ge(0, z'), z', z'')) :|: z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(2, ge(z1 - 1, z2 - 1)), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(1, ge(z1 - 1, z2 - 1)), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0


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

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

   { div, divisible }
   { ge }
   { plus }
   { p }
   { a }
   { or }
   { if2, if, lcmIter }
   { ifTimes, times }
   { lcm }

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 1 }-> divisible(z', z'') :|: z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> div(z' - 1, z'', z1 - 1) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 1 }-> div(1 + (z' - 1), 1 + (z'' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(div(1 + (z2 - 1), 1 + (z1 - 1), 1 + (z1 - 1)), z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(1 + (z'' - 1), z1, 1 + plus(z'' - 1, z2), z3) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 2 }-> lcmIter(z', z'', 0, ifTimes(ge(0, z'), z', z'')) :|: z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(2, ge(z1 - 1, z2 - 1)), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(1, ge(z1 - 1, z2 - 1)), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {div,divisible}, {ge}, {plus}, {p}, {a}, {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 1 }-> divisible(z', z'') :|: z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> div(z' - 1, z'', z1 - 1) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 1 }-> div(1 + (z' - 1), 1 + (z'' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(div(1 + (z2 - 1), 1 + (z1 - 1), 1 + (z1 - 1)), z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(1 + (z'' - 1), z1, 1 + plus(z'' - 1, z2), z3) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 2 }-> lcmIter(z', z'', 0, ifTimes(ge(0, z'), z', z'')) :|: z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(2, ge(z1 - 1, z2 - 1)), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(1, ge(z1 - 1, z2 - 1)), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {div,divisible}, {ge}, {plus}, {p}, {a}, {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 1 }-> divisible(z', z'') :|: z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> div(z' - 1, z'', z1 - 1) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 1 }-> div(1 + (z' - 1), 1 + (z'' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(div(1 + (z2 - 1), 1 + (z1 - 1), 1 + (z1 - 1)), z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(1 + (z'' - 1), z1, 1 + plus(z'' - 1, z2), z3) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 2 }-> lcmIter(z', z'', 0, ifTimes(ge(0, z'), z', z'')) :|: z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(2, ge(z1 - 1, z2 - 1)), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(1, ge(z1 - 1, z2 - 1)), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {div,divisible}, {ge}, {plus}, {p}, {a}, {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: ?, size: O(1) [2]
  divisible: runtime: ?, size: O(1) [2]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: div
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 3*z'

Computed RUNTIME bound using CoFloCo for: divisible
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 5 + 3*z'

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 1 }-> divisible(z', z'') :|: z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> div(z' - 1, z'', z1 - 1) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 1 }-> div(1 + (z' - 1), 1 + (z'' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(div(1 + (z2 - 1), 1 + (z1 - 1), 1 + (z1 - 1)), z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(1 + (z'' - 1), z1, 1 + plus(z'' - 1, z2), z3) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 2 }-> lcmIter(z', z'', 0, ifTimes(ge(0, z'), z', z'')) :|: z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(2, ge(z1 - 1, z2 - 1)), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(1, ge(z1 - 1, z2 - 1)), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {ge}, {plus}, {p}, {a}, {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(1 + (z'' - 1), z1, 1 + plus(z'' - 1, z2), z3) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 2 }-> lcmIter(z', z'', 0, ifTimes(ge(0, z'), z', z'')) :|: z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(2, ge(z1 - 1, z2 - 1)), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(1, ge(z1 - 1, z2 - 1)), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {ge}, {plus}, {p}, {a}, {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]

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

(29) 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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(1 + (z'' - 1), z1, 1 + plus(z'' - 1, z2), z3) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 2 }-> lcmIter(z', z'', 0, ifTimes(ge(0, z'), z', z'')) :|: z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(2, ge(z1 - 1, z2 - 1)), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(1, ge(z1 - 1, z2 - 1)), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {ge}, {plus}, {p}, {a}, {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: ?, size: O(1) [2]

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

(31) 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''

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(1 + (z'' - 1), z1, 1 + plus(z'' - 1, z2), z3) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 2 }-> lcmIter(z', z'', 0, ifTimes(ge(0, z'), z', z'')) :|: z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(2, ge(z1 - 1, z2 - 1)), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(or(1, ge(z1 - 1, z2 - 1)), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {plus}, {p}, {a}, {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z'' }-> s3 :|: s3 >= 0, s3 <= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(1 + (z'' - 1), z1, 1 + plus(z'' - 1, z2), z3) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 4 + z' }-> lcmIter(z', z'', 0, ifTimes(s2, z', z'')) :|: s2 >= 0, s2 <= 2, z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(2, s4), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s4 >= 0, s4 <= 2, z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(1, s5), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s5 >= 0, s5 <= 2, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {plus}, {p}, {a}, {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]

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

(35) 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''

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z'' }-> s3 :|: s3 >= 0, s3 <= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(1 + (z'' - 1), z1, 1 + plus(z'' - 1, z2), z3) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 4 + z' }-> lcmIter(z', z'', 0, ifTimes(s2, z', z'')) :|: s2 >= 0, s2 <= 2, z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(2, s4), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s4 >= 0, s4 <= 2, z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(1, s5), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s5 >= 0, s5 <= 2, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {plus}, {p}, {a}, {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: ?, size: O(n^1) [z' + z'']

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

(37) 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'

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z'' }-> s3 :|: s3 >= 0, s3 <= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(1 + (z'' - 1), z1, 1 + plus(z'' - 1, z2), z3) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 4 + z' }-> lcmIter(z', z'', 0, ifTimes(s2, z', z'')) :|: s2 >= 0, s2 <= 2, z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(2, s4), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s4 >= 0, s4 <= 2, z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(1, s5), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s5 >= 0, s5 <= 2, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {p}, {a}, {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z'' }-> s3 :|: s3 >= 0, s3 <= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 + z'' }-> lcmIter(1 + (z'' - 1), z1, 1 + s6, z3) :|: s6 >= 0, s6 <= z'' - 1 + z2, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 4 + z' }-> lcmIter(z', z'', 0, ifTimes(s2, z', z'')) :|: s2 >= 0, s2 <= 2, z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(2, s4), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s4 >= 0, s4 <= 2, z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(1, s5), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s5 >= 0, s5 <= 2, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {p}, {a}, {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z'' }-> s3 :|: s3 >= 0, s3 <= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 + z'' }-> lcmIter(1 + (z'' - 1), z1, 1 + s6, z3) :|: s6 >= 0, s6 <= z'' - 1 + z2, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 4 + z' }-> lcmIter(z', z'', 0, ifTimes(s2, z', z'')) :|: s2 >= 0, s2 <= 2, z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(2, s4), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s4 >= 0, s4 <= 2, z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(1, s5), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s5 >= 0, s5 <= 2, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {p}, {a}, {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: ?, size: O(n^1) [2 + z']

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z'' }-> s3 :|: s3 >= 0, s3 <= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 + z'' }-> lcmIter(1 + (z'' - 1), z1, 1 + s6, z3) :|: s6 >= 0, s6 <= z'' - 1 + z2, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 4 + z' }-> lcmIter(z', z'', 0, ifTimes(s2, z', z'')) :|: s2 >= 0, s2 <= 2, z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(2, s4), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s4 >= 0, s4 <= 2, z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(1, s5), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s5 >= 0, s5 <= 2, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {a}, {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [2 + z']

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z'' }-> s3 :|: s3 >= 0, s3 <= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 + z'' }-> lcmIter(1 + (z'' - 1), z1, 1 + s6, z3) :|: s6 >= 0, s6 <= z'' - 1 + z2, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 4 + z' }-> lcmIter(z', z'', 0, ifTimes(s2, z', z'')) :|: s2 >= 0, s2 <= 2, z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(2, s4), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s4 >= 0, s4 <= 2, z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(1, s5), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s5 >= 0, s5 <= 2, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {a}, {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [2 + z']

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

(47) 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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z'' }-> s3 :|: s3 >= 0, s3 <= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 + z'' }-> lcmIter(1 + (z'' - 1), z1, 1 + s6, z3) :|: s6 >= 0, s6 <= z'' - 1 + z2, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 4 + z' }-> lcmIter(z', z'', 0, ifTimes(s2, z', z'')) :|: s2 >= 0, s2 <= 2, z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(2, s4), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s4 >= 0, s4 <= 2, z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(1, s5), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s5 >= 0, s5 <= 2, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {a}, {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [2 + z']
  a: runtime: ?, size: O(1) [1]

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

(49) 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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z'' }-> s3 :|: s3 >= 0, s3 <= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 + z'' }-> lcmIter(1 + (z'' - 1), z1, 1 + s6, z3) :|: s6 >= 0, s6 <= z'' - 1 + z2, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 4 + z' }-> lcmIter(z', z'', 0, ifTimes(s2, z', z'')) :|: s2 >= 0, s2 <= 2, z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(2, s4), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s4 >= 0, s4 <= 2, z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(1, s5), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s5 >= 0, s5 <= 2, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [2 + z']
  a: runtime: O(1) [1], size: O(1) [1]

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z'' }-> s3 :|: s3 >= 0, s3 <= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 + z'' }-> lcmIter(1 + (z'' - 1), z1, 1 + s6, z3) :|: s6 >= 0, s6 <= z'' - 1 + z2, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 4 + z' }-> lcmIter(z', z'', 0, ifTimes(s2, z', z'')) :|: s2 >= 0, s2 <= 2, z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(2, s4), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s4 >= 0, s4 <= 2, z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(1, s5), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s5 >= 0, s5 <= 2, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [2 + z']
  a: runtime: O(1) [1], size: O(1) [1]

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z'' }-> s3 :|: s3 >= 0, s3 <= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 + z'' }-> lcmIter(1 + (z'' - 1), z1, 1 + s6, z3) :|: s6 >= 0, s6 <= z'' - 1 + z2, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 4 + z' }-> lcmIter(z', z'', 0, ifTimes(s2, z', z'')) :|: s2 >= 0, s2 <= 2, z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(2, s4), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s4 >= 0, s4 <= 2, z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(1, s5), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s5 >= 0, s5 <= 2, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {or}, {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [2 + z']
  a: runtime: O(1) [1], size: O(1) [1]
  or: runtime: ?, size: O(n^1) [2 + z'']

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z'' }-> s3 :|: s3 >= 0, s3 <= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 + z'' }-> lcmIter(1 + (z'' - 1), z1, 1 + s6, z3) :|: s6 >= 0, s6 <= z'' - 1 + z2, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 4 + z' }-> lcmIter(z', z'', 0, ifTimes(s2, z', z'')) :|: s2 >= 0, s2 <= 2, z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(2, s4), 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s4 >= 0, s4 <= 2, z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 4 + z2 }-> if(or(1, s5), 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s5 >= 0, s5 <= 2, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [2 + z']
  a: runtime: O(1) [1], size: O(1) [1]
  or: runtime: O(1) [1], size: O(n^1) [2 + z'']

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z'' }-> s3 :|: s3 >= 0, s3 <= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 + z'' }-> lcmIter(1 + (z'' - 1), z1, 1 + s6, z3) :|: s6 >= 0, s6 <= z'' - 1 + z2, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 4 + z' }-> lcmIter(z', z'', 0, ifTimes(s2, z', z'')) :|: s2 >= 0, s2 <= 2, z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 5 + z2 }-> if(s8, 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s8 >= 0, s8 <= s4 + 2, s4 >= 0, s4 <= 2, z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 5 + z2 }-> if(s9, 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s9 >= 0, s9 <= s5 + 2, s5 >= 0, s5 <= 2, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [2 + z']
  a: runtime: O(1) [1], size: O(1) [1]
  or: runtime: O(1) [1], size: O(n^1) [2 + z'']

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

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

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

Computed SIZE bound using CoFloCo for: lcmIter
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 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z'' }-> s3 :|: s3 >= 0, s3 <= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 + z'' }-> lcmIter(1 + (z'' - 1), z1, 1 + s6, z3) :|: s6 >= 0, s6 <= z'' - 1 + z2, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 4 + z' }-> lcmIter(z', z'', 0, ifTimes(s2, z', z'')) :|: s2 >= 0, s2 <= 2, z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 5 + z2 }-> if(s8, 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s8 >= 0, s8 <= s4 + 2, s4 >= 0, s4 <= 2, z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 5 + z2 }-> if(s9, 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s9 >= 0, s9 <= s5 + 2, s5 >= 0, s5 <= 2, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [2 + z']
  a: runtime: O(1) [1], size: O(1) [1]
  or: runtime: O(1) [1], size: O(n^1) [2 + z'']
  if2: runtime: ?, size: INF
  if: runtime: ?, size: INF
  lcmIter: runtime: ?, size: INF

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 6 + 3*z' }-> s'' :|: s'' >= 0, s'' <= 2, z1 = 0, z' >= 0, z'' >= 0
   div(z', z'', z1) -{ 2 + 3*z' }-> s1 :|: s1 >= 0, s1 <= 2, z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   divisible(z', z'') -{ 5 + 3*z' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   divisible(z', z'') -{ 1 }-> 2 :|: z'' - 1 >= 0, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z'' }-> s3 :|: s3 >= 0, s3 <= 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
   if(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 6 + 3*z2 }-> if2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1), z3) :|: s >= 0, s <= 2, z1 - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 1, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 2 }-> if2(2, z'', 1 + (z1 - 1), 0, z3) :|: z2 = 0, z'' >= 0, z' = 1, z1 - 1 >= 0, z3 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(0, z'', z1, z2, z3) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 }-> lcmIter(0, z1, z2, z3) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1, z3 >= 0
   if2(z', z'', z1, z2, z3) -{ 2 + z'' }-> lcmIter(1 + (z'' - 1), z1, 1 + s6, z3) :|: s6 >= 0, s6 <= z'' - 1 + z2, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0, z3 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, z'' - 1)) :|: z1 >= 0, z' = 1, z'' - 1 >= 0
   ifTimes(z', z'', z1) -{ 2 }-> plus(z1, times(z1, 1 + (1 + 0))) :|: z'' = 0, z1 >= 0, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z' = 2, z'' >= 0, z1 >= 0
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   lcm(z', z'') -{ 4 + z' }-> lcmIter(z', z'', 0, ifTimes(s2, z', z'')) :|: s2 >= 0, s2 <= 2, z' >= 0, z'' >= 0
   lcmIter(z', z'', z1, z2) -{ 5 + z2 }-> if(s8, 0, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s8 >= 0, s8 <= s4 + 2, s4 >= 0, s4 <= 2, z2 - 1 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   lcmIter(z', z'', z1, z2) -{ 5 + z2 }-> if(s9, 1 + (z' - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s9 >= 0, s9 <= s5 + 2, s5 >= 0, s5 <= 2, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 - 1 >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, 2 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(y'', 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, 1 = y'', y'' >= 0, 1 = 1
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, 2 = 2, 2 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 4 }-> if(2, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, 2 = 2, 1 = y', y' >= 0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 0, z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z2 - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', z1, 0) :|: z1 >= 0, z2 = 0, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
   lcmIter(z', z'', z1, z2) -{ 3 }-> if(0, 1 + (z' - 1), z'', 0, 1 + (z2 - 1)) :|: z1 = 0, z'' >= 0, z' - 1 >= 0, z2 - 1 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
   or(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> ifTimes(2, 0, z'') :|: z'' >= 0, z' = 0
   times(z', z'') -{ 2 }-> ifTimes(1, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0

Function symbols to be analyzed: {if2,if,lcmIter}, {ifTimes,times}, {lcm}
Previous analysis results are:
  div: runtime: O(n^1) [4 + 3*z'], size: O(1) [2]
  divisible: runtime: O(n^1) [5 + 3*z'], size: O(1) [2]
  ge: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [2 + z']
  a: runtime: O(1) [1], size: O(1) [1]
  or: runtime: O(1) [1], size: O(n^1) [2 + z'']
  if2: runtime: INF, size: INF
  if: runtime: ?, size: INF
  lcmIter: runtime: ?, size: INF

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

(63) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   divisible(v0, v1) -> null_divisible [0]
   if(v0, v1, v2, v3, v4) -> null_if [0]
   if2(v0, v1, v2, v3, v4) -> null_if2 [0]
   ifTimes(v0, v1, v2) -> null_ifTimes [0]
   or(v0, v1) -> null_or [0]
   plus(v0, v1) -> null_plus [0]
   p(v0) -> null_p [0]
   ge(v0, v1) -> null_ge [0]
   div(v0, v1, v2) -> null_div [0]

And the following fresh constants:    null_divisible, null_if, null_if2, null_ifTimes, null_or, null_plus, null_p, null_ge, null_div

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

(64)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

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

   lcm(x, y) -> lcmIter(x, y, 0, times(x, y)) [1]
   lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u) [1]
   if(true, x, y, z, u) -> z [1]
   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u) [1]
   if2(true, x, y, z, u) -> z [1]
   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(x, y) -> ifTimes(ge(0, x), x, y) [1]
   ifTimes(true, x, y) -> 0 [1]
   ifTimes(false, x, y) -> plus(y, times(y, p(x))) [1]
   p(s(x)) -> x [1]
   p(0) -> s(s(0)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   or(true, y) -> true [1]
   or(false, y) -> y [1]
   divisible(0, s(y)) -> true [1]
   divisible(s(x), s(y)) -> div(s(x), s(y), s(y)) [1]
   div(x, y, 0) -> divisible(x, y) [1]
   div(0, y, s(z)) -> false [1]
   div(s(x), y, s(z)) -> div(x, y, z) [1]
   a -> b [1]
   a -> c [1]
   divisible(v0, v1) -> null_divisible [0]
   if(v0, v1, v2, v3, v4) -> null_if [0]
   if2(v0, v1, v2, v3, v4) -> null_if2 [0]
   ifTimes(v0, v1, v2) -> null_ifTimes [0]
   or(v0, v1) -> null_or [0]
   plus(v0, v1) -> null_plus [0]
   p(v0) -> null_p [0]
   ge(v0, v1) -> null_ge [0]
   div(v0, v1, v2) -> null_div [0]

The TRS has the following type information:
lcm :: 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p
lcmIter :: 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p
0 :: 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p
times :: 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p
if :: true:false:null_divisible:null_or:null_ge:null_div -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p
or :: true:false:null_divisible:null_or:null_ge:null_div -> true:false:null_divisible:null_or:null_ge:null_div -> true:false:null_divisible:null_or:null_ge:null_div
ge :: 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> true:false:null_divisible:null_or:null_ge:null_div
true :: true:false:null_divisible:null_or:null_ge:null_div
false :: true:false:null_divisible:null_or:null_ge:null_div
if2 :: true:false:null_divisible:null_or:null_ge:null_div -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p
divisible :: 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> true:false:null_divisible:null_or:null_ge:null_div
plus :: 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p
s :: 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p
ifTimes :: true:false:null_divisible:null_or:null_ge:null_div -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p
p :: 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p
div :: 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p -> true:false:null_divisible:null_or:null_ge:null_div
a :: b:c
b :: b:c
c :: b:c
null_divisible :: true:false:null_divisible:null_or:null_ge:null_div
null_if :: 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p
null_if2 :: 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p
null_ifTimes :: 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p
null_or :: true:false:null_divisible:null_or:null_ge:null_div
null_plus :: 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p
null_p :: 0:s:null_if:null_if2:null_ifTimes:null_plus:null_p
null_ge :: true:false:null_divisible:null_or:null_ge:null_div
null_div :: true:false:null_divisible:null_or:null_ge:null_div

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

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

   0 => 0
   true => 2
   false => 1
   b => 0
   c => 1
   null_divisible => 0
   null_if => 0
   null_if2 => 0
   null_ifTimes => 0
   null_or => 0
   null_plus => 0
   null_p => 0
   null_ge => 0
   null_div => 0

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   div(z', z'', z1) -{ 1 }-> divisible(x, y) :|: z1 = 0, z' = x, z'' = y, x >= 0, y >= 0
   div(z', z'', z1) -{ 1 }-> div(x, y, z) :|: z' = 1 + x, z >= 0, z'' = y, x >= 0, y >= 0, z1 = 1 + z
   div(z', z'', z1) -{ 1 }-> 1 :|: z >= 0, z'' = y, y >= 0, z1 = 1 + z, z' = 0
   div(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   divisible(z', z'') -{ 1 }-> div(1 + x, 1 + y, 1 + y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   divisible(z', z'') -{ 1 }-> 2 :|: y >= 0, z'' = 1 + y, z' = 0
   divisible(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   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
   if(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
   if(z', z'', z1, z2, z3) -{ 1 }-> if2(divisible(z, y), x, y, z, u) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1, z3 = u, u >= 0
   if(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
   if2(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
   if2(z', z'', z1, z2, z3) -{ 1 }-> lcmIter(x, y, plus(x, z), u) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1, z3 = u, u >= 0
   if2(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
   ifTimes(z', z'', z1) -{ 1 }-> plus(y, times(y, p(x))) :|: z1 = y, x >= 0, y >= 0, z'' = x, z' = 1
   ifTimes(z', z'', z1) -{ 1 }-> 0 :|: z1 = y, z' = 2, x >= 0, y >= 0, z'' = x
   ifTimes(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   lcm(z', z'') -{ 1 }-> lcmIter(x, y, 0, times(x, y)) :|: z' = x, z'' = y, x >= 0, y >= 0
   lcmIter(z', z'', z1, z2) -{ 1 }-> if(or(ge(0, x), ge(z, u)), x, y, z, u) :|: z1 = z, z2 = u, z >= 0, z' = x, z'' = y, x >= 0, y >= 0, u >= 0
   or(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 1
   or(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' = y, y >= 0
   or(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0
   p(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 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
   times(z', z'') -{ 1 }-> ifTimes(ge(0, x), x, y) :|: z' = x, z'' = y, x >= 0, y >= 0

Only complete derivations are relevant for the runtime complexity.

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

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lcm(z0, z1) -> lcmIter(z0, z1, 0, times(z0, z1))
   lcmIter(z0, z1, z2, z3) -> if(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3)
   if(true, z0, z1, z2, z3) -> z2
   if(false, z0, z1, z2, z3) -> if2(divisible(z2, z1), z0, z1, z2, z3)
   if2(true, z0, z1, z2, z3) -> z2
   if2(false, z0, z1, z2, z3) -> lcmIter(z0, z1, plus(z0, z2), z3)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   or(true, z0) -> true
   or(false, z0) -> z0
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   a -> b
   a -> c
Tuples:
   LCM(z0, z1) -> c1(LCMITER(z0, z1, 0, times(z0, z1)), TIMES(z0, z1))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), OR(ge(0, z0), ge(z2, z3)), GE(0, z0))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), OR(ge(0, z0), ge(z2, z3)), GE(z2, z3))
   IF(true, z0, z1, z2, z3) -> c4
   IF(false, z0, z1, z2, z3) -> c5(IF2(divisible(z2, z1), z0, z1, z2, z3), DIVISIBLE(z2, z1))
   IF2(true, z0, z1, z2, z3) -> c6
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(0, z0) -> c8
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1), GE(0, z0))
   IFTIMES(true, z0, z1) -> c11
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)), P(z0))
   P(s(z0)) -> c13
   P(0) -> c14
   GE(z0, 0) -> c15
   GE(0, s(z0)) -> c16
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   OR(true, z0) -> c18
   OR(false, z0) -> c19
   DIVISIBLE(0, s(z0)) -> c20
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(0, z0, s(z1)) -> c23
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   A -> c25
   A -> c26
S tuples:
   LCM(z0, z1) -> c1(LCMITER(z0, z1, 0, times(z0, z1)), TIMES(z0, z1))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), OR(ge(0, z0), ge(z2, z3)), GE(0, z0))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), OR(ge(0, z0), ge(z2, z3)), GE(z2, z3))
   IF(true, z0, z1, z2, z3) -> c4
   IF(false, z0, z1, z2, z3) -> c5(IF2(divisible(z2, z1), z0, z1, z2, z3), DIVISIBLE(z2, z1))
   IF2(true, z0, z1, z2, z3) -> c6
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(0, z0) -> c8
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1), GE(0, z0))
   IFTIMES(true, z0, z1) -> c11
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)), P(z0))
   P(s(z0)) -> c13
   P(0) -> c14
   GE(z0, 0) -> c15
   GE(0, s(z0)) -> c16
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   OR(true, z0) -> c18
   OR(false, z0) -> c19
   DIVISIBLE(0, s(z0)) -> c20
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(0, z0, s(z1)) -> c23
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   A -> c25
   A -> c26
K tuples:none
Defined Rule Symbols:   lcm_2, lcmIter_4, if_5, if2_5, plus_2, times_2, ifTimes_3, p_1, ge_2, or_2, divisible_2, div_3, a

Defined Pair Symbols:   LCM_2, LCMITER_4, IF_5, IF2_5, PLUS_2, TIMES_2, IFTIMES_3, P_1, GE_2, OR_2, DIVISIBLE_2, DIV_3, A

Compound Symbols:   c1_2, c2_3, c3_3, c4, c5_2, c6, c7_2, c8, c9_1, c10_2, c11, c12_3, c13, c14, c15, c16, c17_1, c18, c19, c20, c21_1, c22_1, c23, c24_1, c25, c26


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

(69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 14 trailing nodes:
   GE(0, s(z0)) -> c16
   IF(true, z0, z1, z2, z3) -> c4
   A -> c25
   A -> c26
   IFTIMES(true, z0, z1) -> c11
   OR(true, z0) -> c18
   IF2(true, z0, z1, z2, z3) -> c6
   P(s(z0)) -> c13
   DIVISIBLE(0, s(z0)) -> c20
   PLUS(0, z0) -> c8
   DIV(0, z0, s(z1)) -> c23
   GE(z0, 0) -> c15
   OR(false, z0) -> c19
   P(0) -> c14

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lcm(z0, z1) -> lcmIter(z0, z1, 0, times(z0, z1))
   lcmIter(z0, z1, z2, z3) -> if(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3)
   if(true, z0, z1, z2, z3) -> z2
   if(false, z0, z1, z2, z3) -> if2(divisible(z2, z1), z0, z1, z2, z3)
   if2(true, z0, z1, z2, z3) -> z2
   if2(false, z0, z1, z2, z3) -> lcmIter(z0, z1, plus(z0, z2), z3)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   or(true, z0) -> true
   or(false, z0) -> z0
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   a -> b
   a -> c
Tuples:
   LCM(z0, z1) -> c1(LCMITER(z0, z1, 0, times(z0, z1)), TIMES(z0, z1))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), OR(ge(0, z0), ge(z2, z3)), GE(0, z0))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), OR(ge(0, z0), ge(z2, z3)), GE(z2, z3))
   IF(false, z0, z1, z2, z3) -> c5(IF2(divisible(z2, z1), z0, z1, z2, z3), DIVISIBLE(z2, z1))
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1), GE(0, z0))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)), P(z0))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
S tuples:
   LCM(z0, z1) -> c1(LCMITER(z0, z1, 0, times(z0, z1)), TIMES(z0, z1))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), OR(ge(0, z0), ge(z2, z3)), GE(0, z0))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), OR(ge(0, z0), ge(z2, z3)), GE(z2, z3))
   IF(false, z0, z1, z2, z3) -> c5(IF2(divisible(z2, z1), z0, z1, z2, z3), DIVISIBLE(z2, z1))
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1), GE(0, z0))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)), P(z0))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
K tuples:none
Defined Rule Symbols:   lcm_2, lcmIter_4, if_5, if2_5, plus_2, times_2, ifTimes_3, p_1, ge_2, or_2, divisible_2, div_3, a

Defined Pair Symbols:   LCM_2, LCMITER_4, IF_5, IF2_5, PLUS_2, TIMES_2, IFTIMES_3, GE_2, DIVISIBLE_2, DIV_3

Compound Symbols:   c1_2, c2_3, c3_3, c5_2, c7_2, c9_1, c10_2, c12_3, c17_1, c21_1, c22_1, c24_1


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

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lcm(z0, z1) -> lcmIter(z0, z1, 0, times(z0, z1))
   lcmIter(z0, z1, z2, z3) -> if(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3)
   if(true, z0, z1, z2, z3) -> z2
   if(false, z0, z1, z2, z3) -> if2(divisible(z2, z1), z0, z1, z2, z3)
   if2(true, z0, z1, z2, z3) -> z2
   if2(false, z0, z1, z2, z3) -> lcmIter(z0, z1, plus(z0, z2), z3)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   or(true, z0) -> true
   or(false, z0) -> z0
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   a -> b
   a -> c
Tuples:
   LCM(z0, z1) -> c1(LCMITER(z0, z1, 0, times(z0, z1)), TIMES(z0, z1))
   IF(false, z0, z1, z2, z3) -> c5(IF2(divisible(z2, z1), z0, z1, z2, z3), DIVISIBLE(z2, z1))
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
S tuples:
   LCM(z0, z1) -> c1(LCMITER(z0, z1, 0, times(z0, z1)), TIMES(z0, z1))
   IF(false, z0, z1, z2, z3) -> c5(IF2(divisible(z2, z1), z0, z1, z2, z3), DIVISIBLE(z2, z1))
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
K tuples:none
Defined Rule Symbols:   lcm_2, lcmIter_4, if_5, if2_5, plus_2, times_2, ifTimes_3, p_1, ge_2, or_2, divisible_2, div_3, a

Defined Pair Symbols:   LCM_2, IF_5, IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3

Compound Symbols:   c1_2, c5_2, c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c10_1, c12_2


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

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lcm(z0, z1) -> lcmIter(z0, z1, 0, times(z0, z1))
   lcmIter(z0, z1, z2, z3) -> if(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3)
   if(true, z0, z1, z2, z3) -> z2
   if(false, z0, z1, z2, z3) -> if2(divisible(z2, z1), z0, z1, z2, z3)
   if2(true, z0, z1, z2, z3) -> z2
   if2(false, z0, z1, z2, z3) -> lcmIter(z0, z1, plus(z0, z2), z3)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   or(true, z0) -> true
   or(false, z0) -> z0
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   a -> b
   a -> c
Tuples:
   IF(false, z0, z1, z2, z3) -> c5(IF2(divisible(z2, z1), z0, z1, z2, z3), DIVISIBLE(z2, z1))
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCM(z0, z1) -> c4(TIMES(z0, z1))
S tuples:
   IF(false, z0, z1, z2, z3) -> c5(IF2(divisible(z2, z1), z0, z1, z2, z3), DIVISIBLE(z2, z1))
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCM(z0, z1) -> c4(TIMES(z0, z1))
K tuples:none
Defined Rule Symbols:   lcm_2, lcmIter_4, if_5, if2_5, plus_2, times_2, ifTimes_3, p_1, ge_2, or_2, divisible_2, div_3, a

Defined Pair Symbols:   IF_5, IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2

Compound Symbols:   c5_2, c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c10_1, c12_2, c4_1


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

(75) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   LCM(z0, z1) -> c4(TIMES(z0, z1))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lcm(z0, z1) -> lcmIter(z0, z1, 0, times(z0, z1))
   lcmIter(z0, z1, z2, z3) -> if(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3)
   if(true, z0, z1, z2, z3) -> z2
   if(false, z0, z1, z2, z3) -> if2(divisible(z2, z1), z0, z1, z2, z3)
   if2(true, z0, z1, z2, z3) -> z2
   if2(false, z0, z1, z2, z3) -> lcmIter(z0, z1, plus(z0, z2), z3)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   or(true, z0) -> true
   or(false, z0) -> z0
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   a -> b
   a -> c
Tuples:
   IF(false, z0, z1, z2, z3) -> c5(IF2(divisible(z2, z1), z0, z1, z2, z3), DIVISIBLE(z2, z1))
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
S tuples:
   IF(false, z0, z1, z2, z3) -> c5(IF2(divisible(z2, z1), z0, z1, z2, z3), DIVISIBLE(z2, z1))
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
K tuples:none
Defined Rule Symbols:   lcm_2, lcmIter_4, if_5, if2_5, plus_2, times_2, ifTimes_3, p_1, ge_2, or_2, divisible_2, div_3, a

Defined Pair Symbols:   IF_5, IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2

Compound Symbols:   c5_2, c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c10_1, c12_2, c4_1


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

(77) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lcm(z0, z1) -> lcmIter(z0, z1, 0, times(z0, z1))
   lcmIter(z0, z1, z2, z3) -> if(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3)
   if(true, z0, z1, z2, z3) -> z2
   if(false, z0, z1, z2, z3) -> if2(divisible(z2, z1), z0, z1, z2, z3)
   if2(true, z0, z1, z2, z3) -> z2
   if2(false, z0, z1, z2, z3) -> lcmIter(z0, z1, plus(z0, z2), z3)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   or(true, z0) -> true
   or(false, z0) -> z0
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   a -> b
   a -> c
Tuples:
   IF(false, z0, z1, z2, z3) -> c5(IF2(divisible(z2, z1), z0, z1, z2, z3), DIVISIBLE(z2, z1))
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
S tuples:
   IF(false, z0, z1, z2, z3) -> c5(IF2(divisible(z2, z1), z0, z1, z2, z3), DIVISIBLE(z2, z1))
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
Defined Rule Symbols:   lcm_2, lcmIter_4, if_5, if2_5, plus_2, times_2, ifTimes_3, p_1, ge_2, or_2, divisible_2, div_3, a

Defined Pair Symbols:   IF_5, IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2

Compound Symbols:   c5_2, c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c10_1, c12_2, c4_1


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

(79) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   lcm(z0, z1) -> lcmIter(z0, z1, 0, times(z0, z1))
   lcmIter(z0, z1, z2, z3) -> if(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3)
   if(true, z0, z1, z2, z3) -> z2
   if(false, z0, z1, z2, z3) -> if2(divisible(z2, z1), z0, z1, z2, z3)
   if2(true, z0, z1, z2, z3) -> z2
   if2(false, z0, z1, z2, z3) -> lcmIter(z0, z1, plus(z0, z2), z3)
   a -> b
   a -> c

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF(false, z0, z1, z2, z3) -> c5(IF2(divisible(z2, z1), z0, z1, z2, z3), DIVISIBLE(z2, z1))
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
S tuples:
   IF(false, z0, z1, z2, z3) -> c5(IF2(divisible(z2, z1), z0, z1, z2, z3), DIVISIBLE(z2, z1))
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF_5, IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2

Compound Symbols:   c5_2, c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c10_1, c12_2, c4_1


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

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, z0, z1, z2, z3) -> c5(IF2(divisible(z2, z1), z0, z1, z2, z3), DIVISIBLE(z2, z1)) by 
   IF(false, x0, s(z0), 0, x3) -> c5(IF2(true, x0, s(z0), 0, x3), DIVISIBLE(0, s(z0)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z0), 0, x3) -> c5(IF2(true, x0, s(z0), 0, x3), DIVISIBLE(0, s(z0)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   IF(false, x0, s(z0), 0, x3) -> c5(IF2(true, x0, s(z0), 0, x3), DIVISIBLE(0, s(z0)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c10_1, c12_2, c4_1, c5_2


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

(83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IF(false, x0, s(z0), 0, x3) -> c5(IF2(true, x0, s(z0), 0, x3), DIVISIBLE(0, s(z0)))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c10_1, c12_2, c4_1, c5_2


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

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(z0, z1, z2, z3) -> c2(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3)) by 
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, 0, s(z0)) -> c2(IF(or(ge(0, x0), false), x0, x1, 0, s(z0)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, 0, s(z0)) -> c2(IF(or(ge(0, x0), false), x0, x1, 0, s(z0)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, 0, s(z0)) -> c2(IF(or(ge(0, x0), false), x0, x1, 0, s(z0)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c10_1, c12_2, c4_1, c5_2, c2_1


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

(87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LCMITER(x0, x1, 0, s(z0)) -> c2(IF(or(ge(0, x0), false), x0, x1, 0, s(z0)))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c10_1, c12_2, c4_1, c5_2, c2_1


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

(89) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
We considered the (Usable) Rules:
   ge(s(z0), s(z1)) -> ge(z0, z1)
   ge(0, s(z0)) -> false
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
And the Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DIV(x_1, x_2, x_3)) = x_2
   POL(DIVISIBLE(x_1, x_2)) = x_2
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 + x_4
   POL(IFTIMES(x_1, x_2, x_3)) = 0
   POL(LCM(x_1, x_2)) = [1] + x_2
   POL(LCMITER(x_1, x_2, x_3, x_4)) = [1] + x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1] + x_1 + x_2
   POL(divisible(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(ge(x_1, x_2)) = [1]
   POL(ifTimes(x_1, x_2, x_3)) = [1] + x_2 + x_3
   POL(or(x_1, x_2)) = x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = 0
   POL(true) = 0

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c10_1, c12_2, c4_1, c5_2, c2_1


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

(91) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(z0, z1, z2, z3) -> c3(IF(or(ge(0, z0), ge(z2, z3)), z0, z1, z2, z3), GE(z2, z3)) by 
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0), GE(z0, 0))
   LCMITER(x0, x1, 0, s(z0)) -> c3(IF(or(ge(0, x0), false), x0, x1, 0, s(z0)), GE(0, s(z0)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0), GE(z0, 0))
   LCMITER(x0, x1, 0, s(z0)) -> c3(IF(or(ge(0, x0), false), x0, x1, 0, s(z0)), GE(0, s(z0)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0), GE(z0, 0))
   LCMITER(x0, x1, 0, s(z0)) -> c3(IF(or(ge(0, x0), false), x0, x1, 0, s(z0)), GE(0, s(z0)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IFTIMES_3, LCM_2, IF_5, LCMITER_4

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c12_2, c4_1, c5_2, c2_1, c3_2


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

(93) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LCMITER(x0, x1, 0, s(z0)) -> c3(IF(or(ge(0, x0), false), x0, x1, 0, s(z0)), GE(0, s(z0)))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0), GE(z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0), GE(z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IFTIMES_3, LCM_2, IF_5, LCMITER_4

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c12_2, c4_1, c5_2, c2_1, c3_2


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

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IFTIMES_3, LCM_2, IF_5, LCMITER_4

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c12_2, c4_1, c5_2, c2_1, c3_2, c3_1


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

(97) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
We considered the (Usable) Rules:
   ge(s(z0), s(z1)) -> ge(z0, z1)
   ge(0, s(z0)) -> false
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
And the Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DIV(x_1, x_2, x_3)) = x_2
   POL(DIVISIBLE(x_1, x_2)) = x_2
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 + x_4
   POL(IFTIMES(x_1, x_2, x_3)) = 0
   POL(LCM(x_1, x_2)) = [1] + x_2
   POL(LCMITER(x_1, x_2, x_3, x_4)) = [1] + x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1] + x_1 + x_2
   POL(divisible(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(ge(x_1, x_2)) = [1]
   POL(ifTimes(x_1, x_2, x_3)) = [1] + x_2 + x_3
   POL(or(x_1, x_2)) = x_2
   POL(p(x_1)) = [1] + x_1
   POL(plus(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = 0
   POL(true) = 0

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(z0, z1) -> c10(IFTIMES(ge(0, z0), z0, z1))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IFTIMES_3, LCM_2, IF_5, LCMITER_4

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c12_2, c4_1, c5_2, c2_1, c3_2, c3_1


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

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

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(0, x1) -> c10(IFTIMES(true, 0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(0, x1) -> c10(IFTIMES(true, 0, x1))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, IFTIMES_3, LCM_2, IF_5, LCMITER_4, TIMES_2

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c12_2, c4_1, c5_2, c2_1, c3_2, c3_1, c10_1


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

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

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   IFTIMES(false, z0, z1) -> c12(PLUS(z1, times(z1, p(z0))), TIMES(z1, p(z0)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, IFTIMES_3, LCM_2, IF_5, LCMITER_4, TIMES_2

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c12_2, c4_1, c5_2, c2_1, c3_2, c3_1, c10_1


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

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

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, times(x1, s(s(0)))), TIMES(x1, p(0)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, times(x1, s(s(0)))), TIMES(x1, p(0)))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCM_2, IF_5, LCMITER_4, TIMES_2, IFTIMES_3

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c4_1, c5_2, c2_1, c3_2, c3_1, c10_1, c12_2


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

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCM_2, IF_5, LCMITER_4, TIMES_2, IFTIMES_3

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c4_1, c5_2, c2_1, c3_2, c3_1, c10_1, c12_2, c_1


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

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

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCM_2, IF_5, LCMITER_4, TIMES_2, IFTIMES_3

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c4_1, c5_2, c2_1, c3_2, c3_1, c10_1, c12_2, c_1


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

(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1))) by 
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1)))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, IF_5, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c5_2, c2_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1


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

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, x0, s(z1), s(z0), x3) -> c5(IF2(div(s(z0), s(z1), s(z1)), x0, s(z1), s(z0), x3), DIVISIBLE(s(z0), s(z1))) by 
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1


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

(113) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
We considered the (Usable) Rules:none
And the Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DIV(x_1, x_2, x_3)) = 0
   POL(DIVISIBLE(x_1, x_2)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = x_3
   POL(IFTIMES(x_1, x_2, x_3)) = x_1
   POL(LCM(x_1, x_2)) = x_2
   POL(LCMITER(x_1, x_2, x_3, x_4)) = x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   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_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(divisible(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(ifTimes(x_1, x_2, x_3)) = x_2
   POL(or(x_1, x_2)) = [1] + x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = [1]
   POL(times(x_1, x_2)) = [1]
   POL(true) = 0

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1


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

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0)) by 
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1


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(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(x0, x1, s(z0), s(z1)) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1))) by 
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1


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(119) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
We considered the (Usable) Rules:
   ge(s(z0), s(z1)) -> ge(z0, z1)
   ge(0, s(z0)) -> false
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
And the Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DIV(x_1, x_2, x_3)) = 0
   POL(DIVISIBLE(x_1, x_2)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = [1] + x_4
   POL(IFTIMES(x_1, x_2, x_3)) = 0
   POL(LCM(x_1, x_2)) = [1]
   POL(LCMITER(x_1, x_2, x_3, x_4)) = [1]
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   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_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(divisible(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(ge(x_1, x_2)) = [1]
   POL(ifTimes(x_1, x_2, x_3)) = x_2
   POL(or(x_1, x_2)) = x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = [1]
   POL(true) = 0

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1


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

(121) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(0, x1, x2, x3) -> c2(IF(or(true, ge(x2, x3)), 0, x1, x2, x3)) by 
   LCMITER(0, x0, x1, x2) -> c2(IF(true, 0, x0, x1, x2))
   LCMITER(0, x0, z0, 0) -> c2(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(0, x0, 0, s(z0)) -> c2(IF(or(true, false), 0, x0, 0, s(z0)))
   LCMITER(0, x0, s(z0), s(z1)) -> c2(IF(or(true, ge(z0, z1)), 0, x0, s(z0), s(z1)))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c2(IF(true, 0, x0, x1, x2))
   LCMITER(0, x0, 0, s(z0)) -> c2(IF(or(true, false), 0, x0, 0, s(z0)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c2(IF(true, 0, x0, x1, x2))
   LCMITER(0, x0, z0, 0) -> c2(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(0, x0, 0, s(z0)) -> c2(IF(or(true, false), 0, x0, 0, s(z0)))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1


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

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c2
   LCMITER(0, x0, 0, s(z0)) -> c2
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, z0, 0) -> c2(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(0, x0, x1, x2) -> c2
   LCMITER(0, x0, 0, s(z0)) -> c2
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2


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

(125) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LCMITER(0, x0, 0, s(z0)) -> c2
We considered the (Usable) Rules:
   plus(s(z0), z1) -> s(plus(z0, z1))
   plus(0, z0) -> z0
And the Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c2
   LCMITER(0, x0, 0, s(z0)) -> c2
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DIV(x_1, x_2, x_3)) = x_2
   POL(DIVISIBLE(x_1, x_2)) = x_2
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = x_3 + x_4
   POL(IFTIMES(x_1, x_2, x_3)) = x_1
   POL(LCM(x_1, x_2)) = [1] + x_2
   POL(LCMITER(x_1, x_2, x_3, x_4)) = x_2 + x_3
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2) = 0
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   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_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(divisible(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(ifTimes(x_1, x_2, x_3)) = x_2
   POL(or(x_1, x_2)) = [1] + x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = x_2
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = [1]
   POL(true) = 0

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c2
   LCMITER(0, x0, 0, s(z0)) -> c2
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, z0, 0) -> c2(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(0, x0, x1, x2) -> c2
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x0, 0, s(z0)) -> c2
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2


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

(127) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LCMITER(0, x0, x1, x2) -> c2
We considered the (Usable) Rules:none
And the Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c2
   LCMITER(0, x0, 0, s(z0)) -> c2
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DIV(x_1, x_2, x_3)) = x_2
   POL(DIVISIBLE(x_1, x_2)) = x_2
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 + x_4
   POL(IFTIMES(x_1, x_2, x_3)) = x_1
   POL(LCM(x_1, x_2)) = [1] + x_2
   POL(LCMITER(x_1, x_2, x_3, x_4)) = [1] + x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2) = 0
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   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_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(divisible(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(ifTimes(x_1, x_2, x_3)) = x_2
   POL(or(x_1, x_2)) = [1] + x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = [1]
   POL(true) = 0

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c2
   LCMITER(0, x0, 0, s(z0)) -> c2
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, z0, 0) -> c2(IF(or(true, true), 0, x0, z0, 0))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x0, 0, s(z0)) -> c2
   LCMITER(0, x0, x1, x2) -> c2
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2


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

(129) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
We considered the (Usable) Rules:
   ge(s(z0), s(z1)) -> ge(z0, z1)
   ge(0, s(z0)) -> false
   or(false, z0) -> z0
   or(true, z0) -> true
   ge(z0, 0) -> true
And the Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c2
   LCMITER(0, x0, 0, s(z0)) -> c2
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DIV(x_1, x_2, x_3)) = x_2
   POL(DIVISIBLE(x_1, x_2)) = x_2
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 + x_4
   POL(IFTIMES(x_1, x_2, x_3)) = 0
   POL(LCM(x_1, x_2)) = [1] + x_2
   POL(LCMITER(x_1, x_2, x_3, x_4)) = [1] + x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2) = 0
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   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_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(divisible(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(ge(x_1, x_2)) = [1]
   POL(ifTimes(x_1, x_2, x_3)) = x_2
   POL(or(x_1, x_2)) = x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = [1]
   POL(true) = 0

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c2
   LCMITER(0, x0, 0, s(z0)) -> c2
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x0, 0, s(z0)) -> c2
   LCMITER(0, x0, x1, x2) -> c2
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c2_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2


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

(131) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(s(z0), x1, x2, x3) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3)) by 
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c2(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c2(IF(or(false, false), s(x0), x1, 0, s(z0)))
   LCMITER(s(x0), x1, s(z0), s(z1)) -> c2(IF(or(false, ge(z0, z1)), s(x0), x1, s(z0), s(z1)))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c2
   LCMITER(0, x0, 0, s(z0)) -> c2
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, 0, s(z0)) -> c2(IF(or(false, false), s(x0), x1, 0, s(z0)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c2(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c2(IF(or(false, false), s(x0), x1, 0, s(z0)))
K tuples:
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, times(z0, z1)))
   LCMITER(x0, x1, z0, 0) -> c2(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x0, 0, s(z0)) -> c2
   LCMITER(0, x0, x1, x2) -> c2
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c2


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

(133) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   LCMITER(0, x0, 0, s(z0)) -> c2
   LCMITER(0, x0, x1, x2) -> c2

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, 0, s(z0)) -> c2(IF(or(false, false), s(x0), x1, 0, s(z0)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c2(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c2(IF(or(false, false), s(x0), x1, 0, s(z0)))
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1


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

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, 0, s(z0)) -> c2
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c2(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c2
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c2


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(137) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LCMITER(s(x0), x1, 0, s(z0)) -> c2
We considered the (Usable) Rules:
   plus(s(z0), z1) -> s(plus(z0, z1))
   plus(0, z0) -> z0
And the Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, 0, s(z0)) -> c2
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DIV(x_1, x_2, x_3)) = x_2
   POL(DIVISIBLE(x_1, x_2)) = x_2
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = x_3 + x_4
   POL(IFTIMES(x_1, x_2, x_3)) = x_1
   POL(LCM(x_1, x_2)) = [1] + x_2
   POL(LCMITER(x_1, x_2, x_3, x_4)) = x_2 + x_3
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2) = 0
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   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_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(divisible(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(ifTimes(x_1, x_2, x_3)) = x_2
   POL(or(x_1, x_2)) = [1] + x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = x_2
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = [1] + x_1
   POL(true) = 0

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, 0, s(z0)) -> c2
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c2(IF(or(false, true), s(x0), x1, z0, 0))
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c2
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c2


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

(139) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
We considered the (Usable) Rules:
   ge(s(z0), s(z1)) -> ge(z0, z1)
   ge(0, s(z0)) -> false
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
And the Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, 0, s(z0)) -> c2
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DIV(x_1, x_2, x_3)) = x_2
   POL(DIVISIBLE(x_1, x_2)) = x_2
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 + x_4
   POL(IFTIMES(x_1, x_2, x_3)) = 0
   POL(LCM(x_1, x_2)) = [1] + x_2
   POL(LCMITER(x_1, x_2, x_3, x_4)) = [1] + x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2) = 0
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   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_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(divisible(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(ge(x_1, x_2)) = [1]
   POL(ifTimes(x_1, x_2, x_3)) = 0
   POL(or(x_1, x_2)) = x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = [1] + x_1
   POL(true) = 0

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, 0, s(z0)) -> c2
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1)))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c2
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c2


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

(141) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(x0, x1, s(z0), s(z1)) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(z0), s(z1)), GE(s(z0), s(z1))) by 
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, 0, s(z0)) -> c2
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c2
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c2


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

(143) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LCMITER(s(x0), x1, 0, s(z0)) -> c2

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1


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

(145) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
We considered the (Usable) Rules:none
And the Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DIV(x_1, x_2, x_3)) = x_2
   POL(DIVISIBLE(x_1, x_2)) = x_2
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 + x_4
   POL(IFTIMES(x_1, x_2, x_3)) = x_1
   POL(LCM(x_1, x_2)) = [1] + x_2
   POL(LCMITER(x_1, x_2, x_3, x_4)) = [1] + x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   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_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1] + x_1 + x_2
   POL(divisible(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(ifTimes(x_1, x_2, x_3)) = x_2
   POL(or(x_1, x_2)) = [1] + x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = [1]
   POL(true) = 0

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1


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

(147) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
We considered the (Usable) Rules:
   ge(s(z0), s(z1)) -> ge(z0, z1)
   ge(0, s(z0)) -> false
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
And the Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DIV(x_1, x_2, x_3)) = x_2
   POL(DIVISIBLE(x_1, x_2)) = x_2
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 + x_4
   POL(IFTIMES(x_1, x_2, x_3)) = 0
   POL(LCM(x_1, x_2)) = [1] + x_2
   POL(LCMITER(x_1, x_2, x_3, x_4)) = [1] + x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   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_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(divisible(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(ge(x_1, x_2)) = [1]
   POL(ifTimes(x_1, x_2, x_3)) = x_2
   POL(or(x_1, x_2)) = x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = [1] + x_1
   POL(true) = 0

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1


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

(149) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(0, x1, x2, x3) -> c3(IF(or(true, ge(x2, x3)), 0, x1, x2, x3), GE(x2, x3)) by 
   LCMITER(0, x0, x1, x2) -> c3(IF(true, 0, x0, x1, x2), GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0), GE(z0, 0))
   LCMITER(0, x0, 0, s(z0)) -> c3(IF(or(true, false), 0, x0, 0, s(z0)), GE(0, s(z0)))
   LCMITER(0, x0, s(z0), s(z1)) -> c3(IF(or(true, ge(z0, z1)), 0, x0, s(z0), s(z1)), GE(s(z0), s(z1)))

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(IF(true, 0, x0, x1, x2), GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0), GE(z0, 0))
   LCMITER(0, x0, 0, s(z0)) -> c3(IF(or(true, false), 0, x0, 0, s(z0)), GE(0, s(z0)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(IF(true, 0, x0, x1, x2), GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0), GE(z0, 0))
   LCMITER(0, x0, 0, s(z0)) -> c3(IF(or(true, false), 0, x0, 0, s(z0)), GE(0, s(z0)))
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1


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

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(0, x0, 0, s(z0)) -> c3
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(0, x0, 0, s(z0)) -> c3
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3


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

(153) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, 0, s(z0)) -> c3
We considered the (Usable) Rules:none
And the Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(0, x0, 0, s(z0)) -> c3
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DIV(x_1, x_2, x_3)) = x_2
   POL(DIVISIBLE(x_1, x_2)) = x_2
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 + x_4
   POL(IFTIMES(x_1, x_2, x_3)) = x_1
   POL(LCM(x_1, x_2)) = [1] + x_2
   POL(LCMITER(x_1, x_2, x_3, x_4)) = [1] + x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3) = 0
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   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_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(divisible(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(ifTimes(x_1, x_2, x_3)) = x_2
   POL(or(x_1, x_2)) = [1] + x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = [1]
   POL(true) = 0

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(0, x0, 0, s(z0)) -> c3
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, 0, s(z0)) -> c3
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3


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

(155) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
We considered the (Usable) Rules:
   ge(s(z0), s(z1)) -> ge(z0, z1)
   ge(0, s(z0)) -> false
   or(false, z0) -> z0
   or(true, z0) -> true
   ge(z0, 0) -> true
And the Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(0, x0, 0, s(z0)) -> c3
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DIV(x_1, x_2, x_3)) = x_2
   POL(DIVISIBLE(x_1, x_2)) = x_2
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 + x_4
   POL(IFTIMES(x_1, x_2, x_3)) = 0
   POL(LCM(x_1, x_2)) = [1] + x_2
   POL(LCMITER(x_1, x_2, x_3, x_4)) = [1] + x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3) = 0
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   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_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1] + x_1 + x_2
   POL(divisible(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(ge(x_1, x_2)) = [1]
   POL(ifTimes(x_1, x_2, x_3)) = x_2
   POL(or(x_1, x_2)) = x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = [1] + x_1
   POL(true) = 0

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(0, x0, 0, s(z0)) -> c3
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, 0, s(z0)) -> c3
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_2, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3


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

(157) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(s(z0), x1, x2, x3) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, x2, x3), GE(x2, x3)) by 
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0), GE(z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c3(IF(or(false, false), s(x0), x1, 0, s(z0)), GE(0, s(z0)))
   LCMITER(s(x0), x1, s(z0), s(z1)) -> c3(IF(or(false, ge(z0, z1)), s(x0), x1, s(z0), s(z1)), GE(s(z0), s(z1)))

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(0, x0, 0, s(z0)) -> c3
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0), GE(z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c3(IF(or(false, false), s(x0), x1, 0, s(z0)), GE(0, s(z0)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0), GE(z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c3(IF(or(false, false), s(x0), x1, 0, s(z0)), GE(0, s(z0)))
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, 0, s(z0)) -> c3
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3


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

(159) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LCMITER(0, x0, 0, s(z0)) -> c3

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0), GE(z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c3(IF(or(false, false), s(x0), x1, 0, s(z0)), GE(0, s(z0)))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0), GE(z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c3(IF(or(false, false), s(x0), x1, 0, s(z0)), GE(0, s(z0)))
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3_2


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

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

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c3
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c3
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3


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

(163) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LCMITER(s(x0), x1, 0, s(z0)) -> c3
We considered the (Usable) Rules:
   plus(s(z0), z1) -> s(plus(z0, z1))
   plus(0, z0) -> z0
And the Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c3
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DIV(x_1, x_2, x_3)) = x_2
   POL(DIVISIBLE(x_1, x_2)) = x_2
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = x_3 + x_4
   POL(IFTIMES(x_1, x_2, x_3)) = x_1
   POL(LCM(x_1, x_2)) = [1] + x_2
   POL(LCMITER(x_1, x_2, x_3, x_4)) = x_2 + x_3
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3) = 0
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   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_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1] + x_1 + x_2
   POL(divisible(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(ifTimes(x_1, x_2, x_3)) = 0
   POL(or(x_1, x_2)) = [1] + x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = x_2
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = [1]
   POL(true) = 0

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c3
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c3
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3


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

(165) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
We considered the (Usable) Rules:
   ge(s(z0), s(z1)) -> ge(z0, z1)
   ge(0, s(z0)) -> false
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
And the Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c3
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DIV(x_1, x_2, x_3)) = x_2
   POL(DIVISIBLE(x_1, x_2)) = x_2
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 + x_4
   POL(IFTIMES(x_1, x_2, x_3)) = 0
   POL(LCM(x_1, x_2)) = [1] + x_2
   POL(LCMITER(x_1, x_2, x_3, x_4)) = [1] + x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3) = 0
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   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_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(divisible(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(ge(x_1, x_2)) = [1]
   POL(ifTimes(x_1, x_2, x_3)) = x_2
   POL(or(x_1, x_2)) = x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = [1]
   POL(true) = 0

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c3
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c3
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, LCMITER_4, TIMES_2, IFTIMES_3, LCM_2, IF_5

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c3_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3


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

(167) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0)) by 
   LCMITER(0, x1, x2, 0) -> c3(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c3(IF(or(false, true), s(z0), x1, x2, 0))

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c3
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
K tuples:
   LCMITER(x0, x1, z0, 0) -> c3(IF(or(ge(0, x0), true), x0, x1, z0, 0))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, 0, s(z0)) -> c3
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IFTIMES_3, LCM_2, IF_5, LCMITER_4

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3_1, c3


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

(169) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LCMITER(s(x0), x1, 0, s(z0)) -> c3

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
S tuples:
   IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   IF2_5, PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IFTIMES_3, LCM_2, IF_5, LCMITER_4

Compound Symbols:   c7_2, c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3_1


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

(171) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF2(false, z0, z1, z2, z3) -> c7(LCMITER(z0, z1, plus(z0, z2), z3), PLUS(z0, z2)) by 
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3), PLUS(0, z0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3), PLUS(0, z0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3), PLUS(0, z0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IFTIMES_3, LCM_2, IF_5, LCMITER_4, IF2_5

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2


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

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0)))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IFTIMES_3, LCM_2, IF_5, LCMITER_4, IF2_5

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1


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

(175) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFTIMES(false, x0, z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, p(x0))), TIMES(z0, p(x0))) by 
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))), TIMES(x1, p(0)))
   IFTIMES(false, x0, 0) -> c12(PLUS(0, ifTimes(true, 0, p(x0))), TIMES(0, p(x0)))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))

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

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))), TIMES(x1, p(0)))
   IFTIMES(false, x0, 0) -> c12(PLUS(0, ifTimes(true, 0, p(x0))), TIMES(0, p(x0)))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))), TIMES(x1, p(0)))
   IFTIMES(false, x0, 0) -> c12(PLUS(0, ifTimes(true, 0, p(x0))), TIMES(0, p(x0)))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IFTIMES_3, LCM_2, IF_5, LCMITER_4, IF2_5

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1


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

(177) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IFTIMES(false, x0, 0) -> c12(PLUS(0, ifTimes(true, 0, p(x0))), TIMES(0, p(x0)))

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

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))), TIMES(x1, p(0)))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))), TIMES(x1, p(0)))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IFTIMES_3, LCM_2, IF_5, LCMITER_4, IF2_5

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1


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

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

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IFTIMES_3, LCM_2, IF_5, LCMITER_4, IF2_5

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c1_1


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

(181) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))

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

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, times(x1, z0)), TIMES(x1, p(s(z0))))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IFTIMES_3, LCM_2, IF_5, LCMITER_4, IF2_5

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c12_2, c_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c1_1


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

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

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

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IFTIMES_3, LCM_2, IF_5, LCMITER_4, IF2_5

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1


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

(185) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0))))) by 
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))

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

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IFTIMES_3, LCM_2, IF_5, LCMITER_4, IF2_5

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1


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

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

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

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   IFTIMES(false, 0, x0) -> c(TIMES(x0, s(s(0))))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IFTIMES(false, 0, x1) -> c(PLUS(x1, times(x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c(TIMES(x1, p(0)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, LCM_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1


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

(189) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IFTIMES(false, 0, x0) -> c(TIMES(x0, s(s(0))))

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

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1)))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, LCM_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c4_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1


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

(191) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCM(z0, z1) -> c4(LCMITER(z0, z1, 0, ifTimes(ge(0, z0), z0, z1))) by 
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))

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

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(193) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0)) by 
   LCMITER(0, x0, x1, 0) -> c2(IF(true, 0, x0, x1, 0))

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

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   LCMITER(0, x0, x1, 0) -> c2(IF(true, 0, x0, x1, 0))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(195) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LCMITER(0, x0, x1, 0) -> c2(IF(true, 0, x0, x1, 0))

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

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, z0, 0) -> c2(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(197) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0)) by 
   LCMITER(s(x0), x1, x2, 0) -> c2(IF(true, s(x0), x1, x2, 0))

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

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, z0, 0) -> c2(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   LCMITER(s(x0), x1, x2, 0) -> c2(IF(true, s(x0), x1, x2, 0))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(199) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LCMITER(s(x0), x1, x2, 0) -> c2(IF(true, s(x0), x1, x2, 0))

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

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, z0, 0) -> c2(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c2(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(201) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(0, x0, z0, 0) -> c2(IF(or(true, true), 0, x0, z0, 0)) by 
   LCMITER(0, x0, x1, 0) -> c2(IF(true, 0, x0, x1, 0))

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

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c2(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   LCMITER(0, x0, x1, 0) -> c2(IF(true, 0, x0, x1, 0))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(0, x1, x2, 0) -> c2(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(203) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LCMITER(0, x0, x1, 0) -> c2(IF(true, 0, x0, x1, 0))

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

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c2(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(205) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(s(x0), x1, z0, 0) -> c2(IF(or(false, true), s(x0), x1, z0, 0)) by 
   LCMITER(s(x0), x1, x2, 0) -> c2(IF(true, s(x0), x1, x2, 0))

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

(206)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   LCMITER(s(x0), x1, x2, 0) -> c2(IF(true, s(x0), x1, x2, 0))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(s(z0), x1, x2, 0) -> c2(IF(or(false, true), s(z0), x1, x2, 0))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(207) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LCMITER(s(x0), x1, x2, 0) -> c2(IF(true, s(x0), x1, x2, 0))

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

(208)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(209) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0)) by 
   LCMITER(0, x0, x1, 0) -> c3(IF(true, 0, x0, x1, 0))

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

(210)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   LCMITER(0, x0, x1, 0) -> c3(IF(true, 0, x0, x1, 0))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(211) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LCMITER(0, x0, x1, 0) -> c3(IF(true, 0, x0, x1, 0))

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

(212)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(0, x1, x2, 0) -> c3(IF(or(true, true), 0, x1, x2, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(213) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0)) by 
   LCMITER(s(x0), x1, x2, 0) -> c3(IF(true, s(x0), x1, x2, 0))

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

(214)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   LCMITER(0, x1, x2, 0) -> c3(IF(or(true, true), 0, x1, x2, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   LCMITER(s(x0), x1, x2, 0) -> c3(IF(true, s(x0), x1, x2, 0))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(215) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LCMITER(s(x0), x1, x2, 0) -> c3(IF(true, s(x0), x1, x2, 0))

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

(216)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(0, x1, x2, 0) -> c3(IF(or(true, true), 0, x1, x2, 0))
   LCMITER(s(z0), x1, x2, 0) -> c3(IF(or(false, true), s(z0), x1, x2, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(217) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(0, x1, x2, 0) -> c3(IF(or(true, true), 0, x1, x2, 0)) by 
   LCMITER(0, x0, x1, 0) -> c3(IF(true, 0, x0, x1, 0))

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

(218)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, 0) -> c3(IF(or(false, true), s(z0), x1, x2, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   LCMITER(0, x0, x1, 0) -> c3(IF(true, 0, x0, x1, 0))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(0, x0, z0, 0) -> c3(IF(or(true, true), 0, x0, z0, 0))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(219) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LCMITER(0, x0, x1, 0) -> c3(IF(true, 0, x0, x1, 0))

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

(220)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   LCMITER(s(z0), x1, x2, 0) -> c3(IF(or(false, true), s(z0), x1, x2, 0))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(221) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LCMITER(s(z0), x1, x2, 0) -> c3(IF(or(false, true), s(z0), x1, x2, 0)) by 
   LCMITER(s(x0), x1, x2, 0) -> c3(IF(true, s(x0), x1, x2, 0))

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

(222)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   LCMITER(s(x0), x1, x2, 0) -> c3(IF(true, s(x0), x1, x2, 0))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, z0, 0) -> c3(IF(or(false, true), s(x0), x1, z0, 0))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(223) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LCMITER(s(x0), x1, x2, 0) -> c3(IF(true, s(x0), x1, x2, 0))

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

(224)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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

(225) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3))) by LCMITER(0, z0, s(z1), s(z2)) -> c2(IF(true, 0, z0, s(z1), s(z2)))
----------------------------------------

(226)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c2(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   LCMITER(0, z0, s(z1), s(z2)) -> c2(IF(true, 0, z0, s(z1), s(z2)))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   LCMITER(0, z0, s(z1), s(z2)) -> c2(IF(true, 0, z0, s(z1), s(z2)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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(227) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LCMITER(0, z0, s(z1), s(z2)) -> c2(IF(true, 0, z0, s(z1), s(z2)))

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

(228)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, s(z0), s(z1)) -> c2(IF(or(true, ge(z0, z1)), 0, x0, s(z0), s(z1)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
S tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   PLUS_2, GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2

Compound Symbols:   c9_1, c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1


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(229) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(z0), z1) -> c9(PLUS(z0, z1)) by 
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))

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

(230)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, s(z0), s(z1)) -> c2(IF(or(true, ge(z0, z1)), 0, x0, s(z0), s(z1)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
S tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2

Compound Symbols:   c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1


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

(231) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3))) by LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
----------------------------------------

(232)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(0, x0, s(z0), s(z1)) -> c2(IF(or(true, ge(z0, z1)), 0, x0, s(z0), s(z1)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
S tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2

Compound Symbols:   c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1


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

(233) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace LCMITER(0, x0, s(z0), s(z1)) -> c2(IF(or(true, ge(z0, z1)), 0, x0, s(z0), s(z1))) by LCMITER(0, z0, s(z1), s(z2)) -> c2(IF(true, 0, z0, s(z1), s(z2)))
----------------------------------------

(234)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c2(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c2(IF(true, 0, z0, s(z1), s(z2)))
S tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2

Compound Symbols:   c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1


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

(235) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LCMITER(0, z0, s(z1), s(z2)) -> c2(IF(true, 0, z0, s(z1), s(z2)))

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

(236)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(s(x0), x1, s(z0), s(z1)) -> c2(IF(or(false, ge(z0, z1)), s(x0), x1, s(z0), s(z1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
S tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2

Compound Symbols:   c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1


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

(237) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace LCMITER(s(x0), x1, s(z0), s(z1)) -> c2(IF(or(false, ge(z0, z1)), s(x0), x1, s(z0), s(z1))) by LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
----------------------------------------

(238)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
S tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2

Compound Symbols:   c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1


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

(239) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3))) by LCMITER(0, z0, s(z1), s(z2)) -> c3(IF(true, 0, z0, s(z1), s(z2)), GE(s(z1), s(z2)))
----------------------------------------

(240)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(IF(true, 0, z0, s(z1), s(z2)), GE(s(z1), s(z2)))
S tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(IF(true, 0, z0, s(z1), s(z2)), GE(s(z1), s(z2)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2

Compound Symbols:   c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1


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

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

(242)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
S tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2

Compound Symbols:   c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1


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

(243) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
We considered the (Usable) Rules:
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(z0, z1, 0) -> divisible(z0, z1)
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(0, z0, s(z1)) -> false
   divisible(0, s(z0)) -> true
And the Tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DIV(x_1, x_2, x_3)) = x_2
   POL(DIVISIBLE(x_1, x_2)) = x_2
   POL(GE(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_3 + x_4
   POL(IFTIMES(x_1, x_2, x_3)) = 0
   POL(LCM(x_1, x_2)) = [1] + x_1 + x_2
   POL(LCMITER(x_1, x_2, x_3, x_4)) = [1] + x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   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(c7(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(div(x_1, x_2, x_3)) = [1]
   POL(divisible(x_1, x_2)) = [1]
   POL(false) = [1]
   POL(ge(x_1, x_2)) = 0
   POL(ifTimes(x_1, x_2, x_3)) = 0
   POL(or(x_1, x_2)) = [1] + x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = [1] + x_1
   POL(true) = 0

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

(244)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
S tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2

Compound Symbols:   c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1


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

(245) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3))) by LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
----------------------------------------

(246)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
S tuples:
   GE(s(z0), s(z1)) -> c17(GE(z0, z1))
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   GE_2, DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2

Compound Symbols:   c17_1, c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1


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

(247) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace GE(s(z0), s(z1)) -> c17(GE(z0, z1)) by 
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))

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

(248)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
S tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))), GE(s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)), GE(s(z0), s(0)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2, GE_2

Compound Symbols:   c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1, c17_1


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

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

(250)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
S tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2, GE_2

Compound Symbols:   c21_1, c22_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1, c17_1


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

(251) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DIV(z0, z1, 0) -> c22(DIVISIBLE(z0, z1)) by 
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))

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

(252)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(0, x1, s(x2), s(x3)) -> c3(IF(or(true, ge(x2, x3)), 0, x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
S tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2, GE_2

Compound Symbols:   c21_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1, c17_1, c22_1


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

(253) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace LCMITER(0, x0, s(z0), s(z1)) -> c3(IF(or(true, ge(z0, z1)), 0, x0, s(z0), s(z1)), GE(s(z0), s(z1))) by LCMITER(0, z0, s(z1), s(z2)) -> c3(IF(true, 0, z0, s(z1), s(z2)), GE(s(z1), s(z2)))
----------------------------------------

(254)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(IF(true, 0, z0, s(z1), s(z2)), GE(s(z1), s(z2)))
S tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2, GE_2

Compound Symbols:   c21_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1, c17_1, c22_1


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

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

(256)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
S tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2, GE_2

Compound Symbols:   c21_1, c24_1, c10_1, c5_2, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1, c17_1, c22_1


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

(257) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, x0, s(z2), s(z0), x3) -> c5(IF2(div(z0, s(z2), z2), x0, s(z2), s(z0), x3), DIVISIBLE(s(z0), s(z2))) by 
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))

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

(258)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(z0), x1, s(x2), s(x3)) -> c3(IF(or(false, ge(x2, x3)), s(z0), x1, s(x2), s(x3)), GE(s(x2), s(x3)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
S tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2, GE_2

Compound Symbols:   c21_1, c24_1, c10_1, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1, c17_1, c22_1, c5_2


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

(259) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace LCMITER(s(x0), x1, s(z0), s(z1)) -> c3(IF(or(false, ge(z0, z1)), s(x0), x1, s(z0), s(z1)), GE(s(z0), s(z1))) by LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
----------------------------------------

(260)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
S tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2, GE_2

Compound Symbols:   c21_1, c24_1, c10_1, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1, c17_1, c22_1, c5_2


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

(261) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0)))) by IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, ifTimes(ge(0, z1), z1, z0)), TIMES(z1, z0))
----------------------------------------

(262)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, ifTimes(ge(0, z1), z1, z0)), TIMES(z1, z0))
S tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, ifTimes(ge(0, z1), z1, z0)), TIMES(z1, z0))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2, GE_2

Compound Symbols:   c21_1, c24_1, c10_1, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1, c17_1, c22_1, c5_2


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

(263) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFTIMES(false, x0, s(z0)) -> c12(PLUS(s(z0), ifTimes(false, s(z0), p(x0))), TIMES(s(z0), p(x0))) by IFTIMES(false, z0, s(z1)) -> c12(PLUS(s(z1), plus(p(z0), times(p(z0), p(s(z1))))), TIMES(s(z1), p(z0)))
----------------------------------------

(264)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, ifTimes(ge(0, z1), z1, z0)), TIMES(z1, z0))
   IFTIMES(false, z0, s(z1)) -> c12(PLUS(s(z1), plus(p(z0), times(p(z0), p(s(z1))))), TIMES(s(z1), p(z0)))
S tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, ifTimes(ge(0, z1), z1, z0)), TIMES(z1, z0))
   IFTIMES(false, z0, s(z1)) -> c12(PLUS(s(z1), plus(p(z0), times(p(z0), p(s(z1))))), TIMES(s(z1), p(z0)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2, GE_2

Compound Symbols:   c21_1, c24_1, c10_1, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1, c17_1, c22_1, c5_2


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

(265) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0))) by IFTIMES(false, 0, z0) -> c1(TIMES(z0, s(s(0))))
----------------------------------------

(266)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(z0), x1) -> c12(PLUS(x1, ifTimes(ge(0, x1), x1, z0)), TIMES(x1, p(s(z0))))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, ifTimes(ge(0, z1), z1, z0)), TIMES(z1, z0))
   IFTIMES(false, z0, s(z1)) -> c12(PLUS(s(z1), plus(p(z0), times(p(z0), p(s(z1))))), TIMES(s(z1), p(z0)))
   IFTIMES(false, 0, z0) -> c1(TIMES(z0, s(s(0))))
S tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, ifTimes(ge(0, z1), z1, z0)), TIMES(z1, z0))
   IFTIMES(false, z0, s(z1)) -> c12(PLUS(s(z1), plus(p(z0), times(p(z0), p(s(z1))))), TIMES(s(z1), p(z0)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, x1) -> c1(TIMES(x1, p(0)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2, GE_2

Compound Symbols:   c21_1, c24_1, c10_1, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c12_2, c1_1, c12_1, c_1, c4_1, c9_1, c17_1, c22_1, c5_2


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

(267) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IFTIMES(false, 0, z0) -> c1(TIMES(z0, s(s(0))))

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

(268)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, s(z1), z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, z1)), TIMES(z0, p(s(z1))))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, ifTimes(ge(0, z1), z1, z0)), TIMES(z1, z0))
   IFTIMES(false, z0, s(z1)) -> c12(PLUS(s(z1), plus(p(z0), times(p(z0), p(s(z1))))), TIMES(s(z1), p(z0)))
S tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, ifTimes(ge(0, z1), z1, z0)), TIMES(z1, z0))
   IFTIMES(false, z0, s(z1)) -> c12(PLUS(s(z1), plus(p(z0), times(p(z0), p(s(z1))))), TIMES(s(z1), p(z0)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2, GE_2

Compound Symbols:   c21_1, c24_1, c10_1, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c1_1, c12_2, c12_1, c_1, c4_1, c9_1, c17_1, c22_1, c5_2


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

(269) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFTIMES(false, s(z1), z0) -> c12(PLUS(z0, ifTimes(ge(0, z0), z0, z1)), TIMES(z0, p(s(z1)))) by IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, ifTimes(ge(0, z1), z1, z0)), TIMES(z1, z0))
----------------------------------------

(270)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, ifTimes(ge(0, z1), z1, z0)), TIMES(z1, z0))
   IFTIMES(false, z0, s(z1)) -> c12(PLUS(s(z1), plus(p(z0), times(p(z0), p(s(z1))))), TIMES(s(z1), p(z0)))
S tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, ifTimes(ge(0, z1), z1, z0)), TIMES(z1, z0))
   IFTIMES(false, z0, s(z1)) -> c12(PLUS(s(z1), plus(p(z0), times(p(z0), p(s(z1))))), TIMES(s(z1), p(z0)))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2, GE_2

Compound Symbols:   c21_1, c24_1, c10_1, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c1_1, c12_1, c_1, c4_1, c9_1, c17_1, c22_1, c5_2, c12_2


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

(271) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFTIMES(false, s(x0), x1) -> c12(TIMES(x1, p(s(x0)))) by IFTIMES(false, s(z0), z1) -> c12(TIMES(z1, z0))
----------------------------------------

(272)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   divisible(0, s(z0)) -> true
   divisible(s(z0), s(z1)) -> div(s(z0), s(z1), s(z1))
   div(s(z0), z1, s(z2)) -> div(z0, z1, z2)
   div(z0, z1, 0) -> divisible(z0, z1)
   div(0, z0, s(z1)) -> false
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   or(true, z0) -> true
   or(false, z0) -> z0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   times(z0, z1) -> ifTimes(ge(0, z0), z0, z1)
   p(s(z0)) -> z0
   p(0) -> s(s(0))
   ifTimes(true, z0, z1) -> 0
   ifTimes(false, z0, z1) -> plus(z1, times(z1, p(z0)))
Tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   IFTIMES(false, 0, z0) -> c(PLUS(z0, ifTimes(ge(0, z0), z0, s(s(0)))))
   LCM(0, x1) -> c4(LCMITER(0, x1, 0, ifTimes(true, 0, x1)))
   LCM(s(z0), x1) -> c4(LCMITER(s(z0), x1, 0, ifTimes(false, s(z0), x1)))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, ifTimes(ge(0, z1), z1, z0)), TIMES(z1, z0))
   IFTIMES(false, z0, s(z1)) -> c12(PLUS(s(z1), plus(p(z0), times(p(z0), p(s(z1))))), TIMES(s(z1), p(z0)))
   IFTIMES(false, s(z0), z1) -> c12(TIMES(z1, z0))
S tuples:
   DIVISIBLE(s(z0), s(z1)) -> c21(DIV(s(z0), s(z1), s(z1)))
   DIV(s(z0), z1, s(z2)) -> c24(DIV(z0, z1, z2))
   TIMES(s(z0), x1) -> c10(IFTIMES(false, s(z0), x1))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c2(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c2(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c2(IF(ge(x2, x3), s(x0), x1, x2, x3))
   LCMITER(x0, x1, s(s(z0)), s(s(z1))) -> c3(IF(or(ge(0, x0), ge(z0, z1)), x0, x1, s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
   LCMITER(s(x0), x1, x2, x3) -> c3(IF(ge(x2, x3), s(x0), x1, x2, x3), GE(x2, x3))
   IF2(false, s(z0), x1, z1, x3) -> c7(LCMITER(s(z0), x1, s(plus(z0, z1)), x3), PLUS(s(z0), z1))
   IF2(false, 0, x1, z0, x3) -> c7(LCMITER(0, x1, z0, x3))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c2(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)))
   LCMITER(s(z0), z1, s(z2), s(z3)) -> c3(IF(ge(z2, z3), s(z0), z1, s(z2), s(z3)), GE(s(z2), s(z3)))
   GE(s(s(y0)), s(s(y1))) -> c17(GE(s(y0), s(y1)))
   LCMITER(x0, x1, s(0), s(s(z0))) -> c3(IF(or(ge(0, x0), false), x0, x1, s(0), s(s(z0))))
   DIV(s(y0), s(y1), 0) -> c22(DIVISIBLE(s(y0), s(y1)))
   IF(false, x0, s(z1), s(x2), s(0)) -> c5(IF2(div(x2, s(z1), z1), x0, s(z1), s(x2), s(0)), DIVISIBLE(s(x2), s(z1)))
   IF(false, x0, s(z1), s(0), s(s(x2))) -> c5(IF2(div(0, s(z1), z1), x0, s(z1), s(0), s(s(x2))), DIVISIBLE(s(0), s(z1)))
   IF(false, x0, s(z1), s(s(x2)), s(s(x3))) -> c5(IF2(div(s(x2), s(z1), z1), x0, s(z1), s(s(x2)), s(s(x3))), DIVISIBLE(s(s(x2)), s(z1)))
   IF(false, s(x0), s(z1), s(z2), x3) -> c5(IF2(div(z2, s(z1), z1), s(x0), s(z1), s(z2), x3), DIVISIBLE(s(z2), s(z1)))
   IF(false, s(x0), s(z1), s(x2), s(x3)) -> c5(IF2(div(x2, s(z1), z1), s(x0), s(z1), s(x2), s(x3)), DIVISIBLE(s(x2), s(z1)))
   IFTIMES(false, s(z0), z1) -> c12(PLUS(z1, ifTimes(ge(0, z1), z1, z0)), TIMES(z1, z0))
   IFTIMES(false, z0, s(z1)) -> c12(PLUS(s(z1), plus(p(z0), times(p(z0), p(s(z1))))), TIMES(s(z1), p(z0)))
   IFTIMES(false, s(z0), z1) -> c12(TIMES(z1, z0))
K tuples:
   IF(false, x0, s(x1), s(x2), x3) -> c5(DIVISIBLE(s(x2), s(x1)))
   LCMITER(x0, x1, s(z0), s(0)) -> c2(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
   LCMITER(x0, x1, s(x2), s(x3)) -> c3(GE(s(x2), s(x3)))
   LCMITER(0, x0, x1, x2) -> c3(GE(x1, x2))
   IFTIMES(false, 0, x1) -> c1(PLUS(x1, ifTimes(ge(0, x1), x1, s(s(0)))))
   LCMITER(0, z0, s(z1), s(z2)) -> c3(GE(s(z1), s(z2)))
   LCMITER(x0, x1, s(z0), s(0)) -> c3(IF(or(ge(0, x0), true), x0, x1, s(z0), s(0)))
Defined Rule Symbols:   divisible_2, div_3, plus_2, or_2, ge_2, times_2, p_1, ifTimes_3

Defined Pair Symbols:   DIVISIBLE_2, DIV_3, TIMES_2, IF_5, LCMITER_4, IF2_5, IFTIMES_3, LCM_2, PLUS_2, GE_2

Compound Symbols:   c21_1, c24_1, c10_1, c5_1, c2_1, c3_2, c3_1, c7_2, c7_1, c1_1, c_1, c4_1, c9_1, c17_1, c22_1, c5_2, c12_2, c12_1