KILLED proof of input_9xGM4Lc9Q7.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 279 ms] (2) CpxRelTRS (3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 19 ms] (16) CpxRNTS (17) InliningProof [UPPER BOUND(ID), 1118 ms] (18) CpxRNTS (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 407 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 139 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 263 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 65 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 373 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 76 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 1071 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 1549 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 304 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 15 ms] (52) CpxRNTS (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 869 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 132 ms] (58) CpxRNTS (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 5087 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 1077 ms] (64) CpxRNTS (65) CompletionProof [UPPER BOUND(ID), 0 ms] (66) CpxTypedWeightedCompleteTrs (67) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 17 ms] (68) CpxRNTS (69) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (70) CdtProblem (71) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (72) CdtProblem (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 140 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms] (80) CdtProblem (81) CdtLeafRemovalProof [ComplexityIfPolyImplication, 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 61 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms] (114) CdtProblem (115) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 1 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 104 ms] (140) CdtProblem (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) @(Nil, ys) -> ys gt0(Cons(x, xs), Nil) -> True gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) gcd(Nil, Nil) -> Nil gcd(Nil, Cons(x, xs)) -> Nil gcd(Cons(x, xs), Nil) -> Nil gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs)) lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs)) eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys)) eqList(Cons(x, xs), Nil) -> False eqList(Nil, Cons(y, ys)) -> False eqList(Nil, Nil) -> True lgth(Nil) -> Nil gt0(Nil, y) -> False monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y) goal(x, y) -> gcd(x, y) The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) monus[Ite](True, Cons(x, xs), y) -> xs gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y) gcd[Ite](True, x, y) -> x gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x)) gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y) Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) @(Nil, ys) -> ys gt0(Cons(x, xs), Nil) -> True gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) gcd(Nil, Nil) -> Nil gcd(Nil, Cons(x, xs)) -> Nil gcd(Cons(x, xs), Nil) -> Nil gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs)) lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs)) eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys)) eqList(Cons(x, xs), Nil) -> False eqList(Nil, Cons(y, ys)) -> False eqList(Nil, Nil) -> True lgth(Nil) -> Nil gt0(Nil, y) -> False monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y) goal(x, y) -> gcd(x, y) The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) monus[Ite](True, Cons(x, xs), y) -> xs gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y) gcd[Ite](True, x, y) -> x gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x)) gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y) Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) @(Nil, ys) -> ys gt0(Cons(x, xs), Nil) -> True gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) gcd(Nil, Nil) -> Nil gcd(Nil, Cons(x, xs)) -> Nil gcd(Cons(x, xs), Nil) -> Nil gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs)) lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs)) eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys)) eqList(Cons(x, xs), Nil) -> False eqList(Nil, Cons(y, ys)) -> False eqList(Nil, Nil) -> True lgth(Nil) -> Nil gt0(Nil, y) -> False monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y) goal(x, y) -> gcd(x, y) The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) monus[Ite](True, Cons(x, xs), y) -> xs gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y) gcd[Ite](True, x, y) -> x gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x)) gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y) Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) 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: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) @(Nil, ys) -> ys gt0(Cons(x, xs), Nil) -> True gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) gcd(Nil, Nil) -> Nil gcd(Nil, Cons(x, xs)) -> Nil gcd(Cons(x, xs), Nil) -> Nil gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs)) lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs)) eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys)) eqList(Cons(x, xs), Nil) -> False eqList(Nil, Cons(y, ys)) -> False eqList(Nil, Nil) -> True lgth(Nil) -> Nil gt0(Nil, y) -> False monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y) goal(x, y) -> gcd(x, y) and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) monus[Ite](True, Cons(x, xs), y) -> xs gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y) gcd[Ite](True, x, y) -> x gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x)) gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (7) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (8) 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: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) [1] @(Nil, ys) -> ys [1] gt0(Cons(x, xs), Nil) -> True [1] gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) [1] gcd(Nil, Nil) -> Nil [1] gcd(Nil, Cons(x, xs)) -> Nil [1] gcd(Cons(x, xs), Nil) -> Nil [1] gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs)) [1] lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs)) [1] eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys)) [1] eqList(Cons(x, xs), Nil) -> False [1] eqList(Nil, Cons(y, ys)) -> False [1] eqList(Nil, Nil) -> True [1] lgth(Nil) -> Nil [1] gt0(Nil, y) -> False [1] monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y) [1] goal(x, y) -> gcd(x, y) [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) [0] monus[Ite](True, Cons(x, xs), y) -> xs [0] gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y) [0] gcd[Ite](True, x, y) -> x [0] gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x)) [0] gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) [1] @(Nil, ys) -> ys [1] gt0(Cons(x, xs), Nil) -> True [1] gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) [1] gcd(Nil, Nil) -> Nil [1] gcd(Nil, Cons(x, xs)) -> Nil [1] gcd(Cons(x, xs), Nil) -> Nil [1] gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs)) [1] lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs)) [1] eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys)) [1] eqList(Cons(x, xs), Nil) -> False [1] eqList(Nil, Cons(y, ys)) -> False [1] eqList(Nil, Nil) -> True [1] lgth(Nil) -> Nil [1] gt0(Nil, y) -> False [1] monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y) [1] goal(x, y) -> gcd(x, y) [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) [0] monus[Ite](True, Cons(x, xs), y) -> xs [0] gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y) [0] gcd[Ite](True, x, y) -> x [0] gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x)) [0] gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y) [0] The TRS has the following type information: @ :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil gt0 :: Cons:Nil -> Cons:Nil -> True:False True :: True:False gcd :: Cons:Nil -> Cons:Nil -> Cons:Nil gcd[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil eqList :: Cons:Nil -> Cons:Nil -> True:False lgth :: Cons:Nil -> Cons:Nil and :: True:False -> True:False -> True:False False :: True:False monus :: Cons:Nil -> Cons:Nil -> Cons:Nil monus[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil gcd[False][Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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: gcd_2 goal_2 (c) The following functions are completely defined: monus_2 gt0_2 lgth_1 eqList_2 @_2 and_2 monus[Ite]_3 gcd[Ite]_3 gcd[False][Ite]_3 Due to the following rules being added: and(v0, v1) -> null_and [0] monus[Ite](v0, v1, v2) -> Nil [0] gcd[Ite](v0, v1, v2) -> Nil [0] gcd[False][Ite](v0, v1, v2) -> Nil [0] And the following fresh constants: null_and ---------------------------------------- (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: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) [1] @(Nil, ys) -> ys [1] gt0(Cons(x, xs), Nil) -> True [1] gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) [1] gcd(Nil, Nil) -> Nil [1] gcd(Nil, Cons(x, xs)) -> Nil [1] gcd(Cons(x, xs), Nil) -> Nil [1] gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs)) [1] lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs)) [1] eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys)) [1] eqList(Cons(x, xs), Nil) -> False [1] eqList(Nil, Cons(y, ys)) -> False [1] eqList(Nil, Nil) -> True [1] lgth(Nil) -> Nil [1] gt0(Nil, y) -> False [1] monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y) [1] goal(x, y) -> gcd(x, y) [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) [0] monus[Ite](True, Cons(x, xs), y) -> xs [0] gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y) [0] gcd[Ite](True, x, y) -> x [0] gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x)) [0] gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y) [0] and(v0, v1) -> null_and [0] monus[Ite](v0, v1, v2) -> Nil [0] gcd[Ite](v0, v1, v2) -> Nil [0] gcd[False][Ite](v0, v1, v2) -> Nil [0] The TRS has the following type information: @ :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil gt0 :: Cons:Nil -> Cons:Nil -> True:False:null_and True :: True:False:null_and gcd :: Cons:Nil -> Cons:Nil -> Cons:Nil gcd[Ite] :: True:False:null_and -> Cons:Nil -> Cons:Nil -> Cons:Nil eqList :: Cons:Nil -> Cons:Nil -> True:False:null_and lgth :: Cons:Nil -> Cons:Nil and :: True:False:null_and -> True:False:null_and -> True:False:null_and False :: True:False:null_and monus :: Cons:Nil -> Cons:Nil -> Cons:Nil monus[Ite] :: True:False:null_and -> Cons:Nil -> Cons:Nil -> Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil gcd[False][Ite] :: True:False:null_and -> Cons:Nil -> Cons:Nil -> Cons:Nil null_and :: True:False:null_and Rewrite Strategy: INNERMOST ---------------------------------------- (13) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (14) 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: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) [1] @(Nil, ys) -> ys [1] gt0(Cons(x, xs), Nil) -> True [1] gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) [1] gcd(Nil, Nil) -> Nil [1] gcd(Nil, Cons(x, xs)) -> Nil [1] gcd(Cons(x, xs), Nil) -> Nil [1] gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), Cons(x', xs'), Cons(x, xs)) [2] lgth(Cons(x, Cons(x'', xs''))) -> @(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(xs''))) [2] lgth(Cons(x, Nil)) -> @(Cons(Nil, Nil), Nil) [2] eqList(Cons(Cons(x1, xs1), Cons(x3, xs3)), Cons(Cons(y', ys'), Cons(y1, ys1))) -> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) [3] eqList(Cons(Cons(x1, xs1), Cons(x4, xs4)), Cons(Cons(y', ys'), Nil)) -> and(and(eqList(x1, y'), eqList(xs1, ys')), False) [3] eqList(Cons(Cons(x1, xs1), Nil), Cons(Cons(y', ys'), Cons(y2, ys2))) -> and(and(eqList(x1, y'), eqList(xs1, ys')), False) [3] eqList(Cons(Cons(x1, xs1), Nil), Cons(Cons(y', ys'), Nil)) -> and(and(eqList(x1, y'), eqList(xs1, ys')), True) [3] eqList(Cons(Cons(x2, xs2), Cons(x5, xs5)), Cons(Nil, Cons(y3, ys3))) -> and(False, and(eqList(x5, y3), eqList(xs5, ys3))) [3] eqList(Cons(Cons(x2, xs2), Cons(x6, xs6)), Cons(Nil, Nil)) -> and(False, False) [3] eqList(Cons(Cons(x2, xs2), Nil), Cons(Nil, Cons(y4, ys4))) -> and(False, False) [3] eqList(Cons(Cons(x2, xs2), Nil), Cons(Nil, Nil)) -> and(False, True) [3] eqList(Cons(Nil, Cons(x7, xs7)), Cons(Cons(y'', ys''), Cons(y5, ys5))) -> and(False, and(eqList(x7, y5), eqList(xs7, ys5))) [3] eqList(Cons(Nil, Cons(x8, xs8)), Cons(Cons(y'', ys''), Nil)) -> and(False, False) [3] eqList(Cons(Nil, Nil), Cons(Cons(y'', ys''), Cons(y6, ys6))) -> and(False, False) [3] eqList(Cons(Nil, Nil), Cons(Cons(y'', ys''), Nil)) -> and(False, True) [3] eqList(Cons(Nil, Cons(x9, xs9)), Cons(Nil, Cons(y7, ys7))) -> and(True, and(eqList(x9, y7), eqList(xs9, ys7))) [3] eqList(Cons(Nil, Cons(x10, xs10)), Cons(Nil, Nil)) -> and(True, False) [3] eqList(Cons(Nil, Nil), Cons(Nil, Cons(y8, ys8))) -> and(True, False) [3] eqList(Cons(Nil, Nil), Cons(Nil, Nil)) -> and(True, True) [3] eqList(Cons(x, xs), Nil) -> False [1] eqList(Nil, Cons(y, ys)) -> False [1] eqList(Nil, Nil) -> True [1] lgth(Nil) -> Nil [1] gt0(Nil, y) -> False [1] monus(x, Cons(x11, xs11)) -> monus[Ite](eqList(@(Cons(Nil, Nil), lgth(xs11)), Cons(Nil, Nil)), x, Cons(x11, xs11)) [2] monus(x, Nil) -> monus[Ite](eqList(Nil, Cons(Nil, Nil)), x, Nil) [2] goal(x, y) -> gcd(x, y) [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) [0] monus[Ite](True, Cons(x, xs), y) -> xs [0] gcd[Ite](False, Cons(x12, xs12), Nil) -> gcd[False][Ite](True, Cons(x12, xs12), Nil) [1] gcd[Ite](False, Cons(x''', xs'''), Cons(x13, xs13)) -> gcd[False][Ite](gt0(xs''', xs13), Cons(x''', xs'''), Cons(x13, xs13)) [1] gcd[Ite](False, Nil, y) -> gcd[False][Ite](False, Nil, y) [1] gcd[Ite](True, x, y) -> x [0] gcd[False][Ite](False, x, y) -> gcd(x, monus[Ite](eqList(lgth(x), Cons(Nil, Nil)), y, x)) [1] gcd[False][Ite](True, x, y) -> gcd(monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y), y) [1] and(v0, v1) -> null_and [0] monus[Ite](v0, v1, v2) -> Nil [0] gcd[Ite](v0, v1, v2) -> Nil [0] gcd[False][Ite](v0, v1, v2) -> Nil [0] The TRS has the following type information: @ :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil gt0 :: Cons:Nil -> Cons:Nil -> True:False:null_and True :: True:False:null_and gcd :: Cons:Nil -> Cons:Nil -> Cons:Nil gcd[Ite] :: True:False:null_and -> Cons:Nil -> Cons:Nil -> Cons:Nil eqList :: Cons:Nil -> Cons:Nil -> True:False:null_and lgth :: Cons:Nil -> Cons:Nil and :: True:False:null_and -> True:False:null_and -> True:False:null_and False :: True:False:null_and monus :: Cons:Nil -> Cons:Nil -> Cons:Nil monus[Ite] :: True:False:null_and -> Cons:Nil -> Cons:Nil -> Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil gcd[False][Ite] :: True:False:null_and -> Cons:Nil -> Cons:Nil -> Cons:Nil null_and :: True:False:null_and Rewrite Strategy: INNERMOST ---------------------------------------- (15) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: Nil => 0 True => 2 False => 1 null_and => 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(2, 2) :|: z = 1 + 0 + 0, z' = 1 + 0 + 0 eqList(z, z') -{ 3 }-> and(2, 1) :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10) eqList(z, z') -{ 3 }-> and(2, 1) :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8) eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 3 }-> and(1, 2) :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, 2) :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0 eqList(z, z') -{ 3 }-> and(1, 1) :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, 1) :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, 1) :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0 eqList(z, z') -{ 3 }-> and(1, 1) :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(x, monus[Ite](eqList(lgth(x), 1 + 0 + 0), y, x)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(y), 1 + 0 + 0), x, y), y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 gcd[Ite](z, z', z'') -{ 0 }-> x :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](gt0(xs''', xs13), 1 + x''' + xs''', 1 + x13 + xs13) :|: xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 goal(z, z') -{ 1 }-> gcd(x, y) :|: x >= 0, y >= 0, z = x, z' = y gt0(z, z') -{ 1 }-> gt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, 0) :|: x >= 0, z = 1 + x + 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), x, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, x >= 0, xs11 >= 0, x11 >= 0, z = x monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), x, 0) :|: x >= 0, z = x, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, z'' = y, x >= 0, y >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 ---------------------------------------- (17) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(x, monus[Ite](eqList(lgth(x), 1 + 0 + 0), y, x)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(y), 1 + 0 + 0), x, y), y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 gcd[Ite](z, z', z'') -{ 0 }-> x :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](gt0(xs''', xs13), 1 + x''' + xs''', 1 + x13 + xs13) :|: xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 goal(z, z') -{ 1 }-> gcd(x, y) :|: x >= 0, y >= 0, z = x, z' = y gt0(z, z') -{ 1 }-> gt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, 0) :|: x >= 0, z = 1 + x + 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), x, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, x >= 0, xs11 >= 0, x11 >= 0, z = x monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), x, 0) :|: x >= 0, z = x, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, z'' = y, x >= 0, y >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 ---------------------------------------- (19) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](gt0(xs''', xs13), 1 + x''' + xs''', 1 + x13 + xs13) :|: xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 1 }-> gt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, 0) :|: z - 1 >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), z, 0) :|: z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 ---------------------------------------- (21) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { gt0 } { and } { @ } { eqList } { lgth } { monus[Ite], monus } { gcd, gcd[Ite], gcd[False][Ite] } { goal } ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](gt0(xs''', xs13), 1 + x''' + xs''', 1 + x13 + xs13) :|: xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 1 }-> gt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, 0) :|: z - 1 >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), z, 0) :|: z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {gt0}, {and}, {@}, {eqList}, {lgth}, {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](gt0(xs''', xs13), 1 + x''' + xs''', 1 + x13 + xs13) :|: xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 1 }-> gt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, 0) :|: z - 1 >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), z, 0) :|: z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {gt0}, {and}, {@}, {eqList}, {lgth}, {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: gt0 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](gt0(xs''', xs13), 1 + x''' + xs''', 1 + x13 + xs13) :|: xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 1 }-> gt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, 0) :|: z - 1 >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), z, 0) :|: z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {gt0}, {and}, {@}, {eqList}, {lgth}, {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: ?, size: O(1) [2] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: gt0 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](gt0(xs''', xs13), 1 + x''' + xs''', 1 + x13 + xs13) :|: xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 1 }-> gt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, 0) :|: z - 1 >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), z, 0) :|: z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {and}, {@}, {eqList}, {lgth}, {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, 0) :|: z - 1 >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), z, 0) :|: z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {and}, {@}, {eqList}, {lgth}, {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, 0) :|: z - 1 >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), z, 0) :|: z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {and}, {@}, {eqList}, {lgth}, {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: ?, size: O(1) [2] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, 0) :|: z - 1 >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), z, 0) :|: z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {@}, {eqList}, {lgth}, {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (35) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, 0) :|: z - 1 >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), z, 0) :|: z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {@}, {eqList}, {lgth}, {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: @ after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, 0) :|: z - 1 >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), z, 0) :|: z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {@}, {eqList}, {lgth}, {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] @: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: @ after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, 0) :|: z - 1 >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), z, 0) :|: z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {eqList}, {lgth}, {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (41) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 2 + xs }-> 1 + x + s'' :|: s'' >= 0, s'' <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + 0 + 0, z - 1 >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), z, 0) :|: z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {eqList}, {lgth}, {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: eqList after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 2 + xs }-> 1 + x + s'' :|: s'' >= 0, s'' <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + 0 + 0, z - 1 >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), z, 0) :|: z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {eqList}, {lgth}, {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] eqList: runtime: ?, size: O(1) [2] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: eqList after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 78 + 156*z' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 2 + xs }-> 1 + x + s'' :|: s'' >= 0, s'' <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), and(eqList(x3, y1), eqList(xs3, ys1))) :|: y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 2) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 3 }-> and(and(eqList(x1, y'), eqList(xs1, ys')), 1) :|: z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 3 }-> and(2, and(eqList(x9, y7), eqList(xs9, ys7))) :|: z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x5, y3), eqList(xs5, ys3))) :|: xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 3 }-> and(1, and(eqList(x7, y5), eqList(xs7, ys5))) :|: y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 2 }-> gcd[Ite](and(eqList(x', x), eqList(xs', xs)), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + 0 + 0, z - 1 >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(0, 1 + 0 + 0), z, 0) :|: z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {lgth}, {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] eqList: runtime: O(n^1) [78 + 156*z'], size: O(1) [2] ---------------------------------------- (47) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 2 + xs }-> 1 + x + s'' :|: s'' >= 0, s'' <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s16 :|: s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s20 :|: s17 >= 0, s17 <= 2, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= 2, s20 >= 0, s20 <= 2, z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s24 :|: s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 159 + 156*y3 + 156*ys3 }-> s28 :|: s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, s27 >= 0, s27 <= 2, s28 >= 0, s28 <= 2, xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 159 + 156*y5 + 156*ys5 }-> s32 :|: s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, s31 >= 0, s31 <= 2, s32 >= 0, s32 <= 2, y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 159 + 156*y7 + 156*ys7 }-> s36 :|: s33 >= 0, s33 <= 2, s34 >= 0, s34 <= 2, s35 >= 0, s35 <= 2, s36 >= 0, s36 <= 2, z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 315 + 156*y' + 156*y1 + 156*ys' + 156*ys1 }-> s8 :|: s2 >= 0, s2 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 158 + 156*x + 156*xs }-> gcd[Ite](s12, 1 + x' + xs', 1 + x + xs) :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + 0 + 0, z - 1 >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 236 }-> monus[Ite](s9, z, 0) :|: s9 >= 0, s9 <= 2, z >= 0, z' = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {lgth}, {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] eqList: runtime: O(n^1) [78 + 156*z'], size: O(1) [2] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: lgth after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 2 + xs }-> 1 + x + s'' :|: s'' >= 0, s'' <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s16 :|: s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s20 :|: s17 >= 0, s17 <= 2, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= 2, s20 >= 0, s20 <= 2, z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s24 :|: s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 159 + 156*y3 + 156*ys3 }-> s28 :|: s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, s27 >= 0, s27 <= 2, s28 >= 0, s28 <= 2, xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 159 + 156*y5 + 156*ys5 }-> s32 :|: s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, s31 >= 0, s31 <= 2, s32 >= 0, s32 <= 2, y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 159 + 156*y7 + 156*ys7 }-> s36 :|: s33 >= 0, s33 <= 2, s34 >= 0, s34 <= 2, s35 >= 0, s35 <= 2, s36 >= 0, s36 <= 2, z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 315 + 156*y' + 156*y1 + 156*ys' + 156*ys1 }-> s8 :|: s2 >= 0, s2 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 158 + 156*x + 156*xs }-> gcd[Ite](s12, 1 + x' + xs', 1 + x + xs) :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + 0 + 0, z - 1 >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 236 }-> monus[Ite](s9, z, 0) :|: s9 >= 0, s9 <= 2, z >= 0, z' = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {lgth}, {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] eqList: runtime: O(n^1) [78 + 156*z'], size: O(1) [2] lgth: runtime: ?, size: O(n^1) [z] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: lgth after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 5 + 6*z ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 2 + xs }-> 1 + x + s'' :|: s'' >= 0, s'' <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s16 :|: s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s20 :|: s17 >= 0, s17 <= 2, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= 2, s20 >= 0, s20 <= 2, z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s24 :|: s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 159 + 156*y3 + 156*ys3 }-> s28 :|: s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, s27 >= 0, s27 <= 2, s28 >= 0, s28 <= 2, xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 159 + 156*y5 + 156*ys5 }-> s32 :|: s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, s31 >= 0, s31 <= 2, s32 >= 0, s32 <= 2, y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 159 + 156*y7 + 156*ys7 }-> s36 :|: s33 >= 0, s33 <= 2, s34 >= 0, s34 <= 2, s35 >= 0, s35 <= 2, s36 >= 0, s36 <= 2, z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 315 + 156*y' + 156*y1 + 156*ys' + 156*ys1 }-> s8 :|: s2 >= 0, s2 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 158 + 156*x + 156*xs }-> gcd[Ite](s12, 1 + x' + xs', 1 + x + xs) :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(z', monus[Ite](eqList(lgth(z'), 1 + 0 + 0), z'', z')) :|: z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 1 }-> gcd(monus[Ite](eqList(lgth(z''), 1 + 0 + 0), z', z''), z'') :|: z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + 0 + 0, z - 1 >= 0 lgth(z) -{ 2 }-> @(1 + 0 + 0, @(1 + 0 + 0, lgth(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 236 }-> monus[Ite](s9, z, 0) :|: s9 >= 0, s9 <= 2, z >= 0, z' = 0 monus(z, z') -{ 2 }-> monus[Ite](eqList(@(1 + 0 + 0, lgth(xs11)), 1 + 0 + 0), z, 1 + x11 + xs11) :|: z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] eqList: runtime: O(n^1) [78 + 156*z'], size: O(1) [2] lgth: runtime: O(n^1) [5 + 6*z], size: O(n^1) [z] ---------------------------------------- (53) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 2 + xs }-> 1 + x + s'' :|: s'' >= 0, s'' <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s16 :|: s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s20 :|: s17 >= 0, s17 <= 2, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= 2, s20 >= 0, s20 <= 2, z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s24 :|: s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 159 + 156*y3 + 156*ys3 }-> s28 :|: s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, s27 >= 0, s27 <= 2, s28 >= 0, s28 <= 2, xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 159 + 156*y5 + 156*ys5 }-> s32 :|: s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, s31 >= 0, s31 <= 2, s32 >= 0, s32 <= 2, y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 159 + 156*y7 + 156*ys7 }-> s36 :|: s33 >= 0, s33 <= 2, s34 >= 0, s34 <= 2, s35 >= 0, s35 <= 2, s36 >= 0, s36 <= 2, z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 315 + 156*y' + 156*y1 + 156*ys' + 156*ys1 }-> s8 :|: s2 >= 0, s2 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 158 + 156*x + 156*xs }-> gcd[Ite](s12, 1 + x' + xs', 1 + x + xs) :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 240 + 6*z' }-> gcd(z', monus[Ite](s44, z'', z')) :|: s43 >= 0, s43 <= z', s44 >= 0, s44 <= 2, z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 240 + 6*z'' }-> gcd(monus[Ite](s46, z', z''), z'') :|: s45 >= 0, s45 <= z'', s46 >= 0, s46 <= 2, z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + 0 + 0, z - 1 >= 0 lgth(z) -{ 11 + 6*xs'' }-> s39 :|: s37 >= 0, s37 <= xs'', s38 >= 0, s38 <= 1 + 0 + 0 + s37, s39 >= 0, s39 <= 1 + 0 + 0 + s38, z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 243 + 6*xs11 }-> monus[Ite](s42, z, 1 + x11 + xs11) :|: s40 >= 0, s40 <= xs11, s41 >= 0, s41 <= 1 + 0 + 0 + s40, s42 >= 0, s42 <= 2, z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 236 }-> monus[Ite](s9, z, 0) :|: s9 >= 0, s9 <= 2, z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] eqList: runtime: O(n^1) [78 + 156*z'], size: O(1) [2] lgth: runtime: O(n^1) [5 + 6*z], size: O(n^1) [z] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: monus[Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' Computed SIZE bound using CoFloCo for: monus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 2 + xs }-> 1 + x + s'' :|: s'' >= 0, s'' <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s16 :|: s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s20 :|: s17 >= 0, s17 <= 2, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= 2, s20 >= 0, s20 <= 2, z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s24 :|: s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 159 + 156*y3 + 156*ys3 }-> s28 :|: s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, s27 >= 0, s27 <= 2, s28 >= 0, s28 <= 2, xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 159 + 156*y5 + 156*ys5 }-> s32 :|: s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, s31 >= 0, s31 <= 2, s32 >= 0, s32 <= 2, y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 159 + 156*y7 + 156*ys7 }-> s36 :|: s33 >= 0, s33 <= 2, s34 >= 0, s34 <= 2, s35 >= 0, s35 <= 2, s36 >= 0, s36 <= 2, z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 315 + 156*y' + 156*y1 + 156*ys' + 156*ys1 }-> s8 :|: s2 >= 0, s2 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 158 + 156*x + 156*xs }-> gcd[Ite](s12, 1 + x' + xs', 1 + x + xs) :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 240 + 6*z' }-> gcd(z', monus[Ite](s44, z'', z')) :|: s43 >= 0, s43 <= z', s44 >= 0, s44 <= 2, z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 240 + 6*z'' }-> gcd(monus[Ite](s46, z', z''), z'') :|: s45 >= 0, s45 <= z'', s46 >= 0, s46 <= 2, z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + 0 + 0, z - 1 >= 0 lgth(z) -{ 11 + 6*xs'' }-> s39 :|: s37 >= 0, s37 <= xs'', s38 >= 0, s38 <= 1 + 0 + 0 + s37, s39 >= 0, s39 <= 1 + 0 + 0 + s38, z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 243 + 6*xs11 }-> monus[Ite](s42, z, 1 + x11 + xs11) :|: s40 >= 0, s40 <= xs11, s41 >= 0, s41 <= 1 + 0 + 0 + s40, s42 >= 0, s42 <= 2, z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 236 }-> monus[Ite](s9, z, 0) :|: s9 >= 0, s9 <= 2, z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {monus[Ite],monus}, {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] eqList: runtime: O(n^1) [78 + 156*z'], size: O(1) [2] lgth: runtime: O(n^1) [5 + 6*z], size: O(n^1) [z] monus[Ite]: runtime: ?, size: O(n^1) [z'] monus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: monus[Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 958*z'' + 12*z''^2 Computed RUNTIME bound using KoAT for: monus after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1449 + 1970*z' + 48*z'^2 ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 2 + xs }-> 1 + x + s'' :|: s'' >= 0, s'' <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s16 :|: s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s20 :|: s17 >= 0, s17 <= 2, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= 2, s20 >= 0, s20 <= 2, z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s24 :|: s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 159 + 156*y3 + 156*ys3 }-> s28 :|: s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, s27 >= 0, s27 <= 2, s28 >= 0, s28 <= 2, xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 159 + 156*y5 + 156*ys5 }-> s32 :|: s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, s31 >= 0, s31 <= 2, s32 >= 0, s32 <= 2, y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 159 + 156*y7 + 156*ys7 }-> s36 :|: s33 >= 0, s33 <= 2, s34 >= 0, s34 <= 2, s35 >= 0, s35 <= 2, s36 >= 0, s36 <= 2, z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 315 + 156*y' + 156*y1 + 156*ys' + 156*ys1 }-> s8 :|: s2 >= 0, s2 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 158 + 156*x + 156*xs }-> gcd[Ite](s12, 1 + x' + xs', 1 + x + xs) :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 240 + 6*z' }-> gcd(z', monus[Ite](s44, z'', z')) :|: s43 >= 0, s43 <= z', s44 >= 0, s44 <= 2, z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 240 + 6*z'' }-> gcd(monus[Ite](s46, z', z''), z'') :|: s45 >= 0, s45 <= z'', s46 >= 0, s46 <= 2, z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + 0 + 0, z - 1 >= 0 lgth(z) -{ 11 + 6*xs'' }-> s39 :|: s37 >= 0, s37 <= xs'', s38 >= 0, s38 <= 1 + 0 + 0 + s37, s39 >= 0, s39 <= 1 + 0 + 0 + s38, z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 243 + 6*xs11 }-> monus[Ite](s42, z, 1 + x11 + xs11) :|: s40 >= 0, s40 <= xs11, s41 >= 0, s41 <= 1 + 0 + 0 + s40, s42 >= 0, s42 <= 2, z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 236 }-> monus[Ite](s9, z, 0) :|: s9 >= 0, s9 <= 2, z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] eqList: runtime: O(n^1) [78 + 156*z'], size: O(1) [2] lgth: runtime: O(n^1) [5 + 6*z], size: O(n^1) [z] monus[Ite]: runtime: O(n^2) [958*z'' + 12*z''^2], size: O(n^1) [z'] monus: runtime: O(n^2) [1449 + 1970*z' + 48*z'^2], size: O(n^1) [z] ---------------------------------------- (59) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 2 + xs }-> 1 + x + s'' :|: s'' >= 0, s'' <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s16 :|: s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s20 :|: s17 >= 0, s17 <= 2, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= 2, s20 >= 0, s20 <= 2, z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s24 :|: s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 159 + 156*y3 + 156*ys3 }-> s28 :|: s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, s27 >= 0, s27 <= 2, s28 >= 0, s28 <= 2, xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 159 + 156*y5 + 156*ys5 }-> s32 :|: s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, s31 >= 0, s31 <= 2, s32 >= 0, s32 <= 2, y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 159 + 156*y7 + 156*ys7 }-> s36 :|: s33 >= 0, s33 <= 2, s34 >= 0, s34 <= 2, s35 >= 0, s35 <= 2, s36 >= 0, s36 <= 2, z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 315 + 156*y' + 156*y1 + 156*ys' + 156*ys1 }-> s8 :|: s2 >= 0, s2 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 158 + 156*x + 156*xs }-> gcd[Ite](s12, 1 + x' + xs', 1 + x + xs) :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 240 + 964*z'' + 12*z''^2 }-> gcd(s51, z'') :|: s51 >= 0, s51 <= z', s45 >= 0, s45 <= z'', s46 >= 0, s46 <= 2, z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 240 + 964*z' + 12*z'^2 }-> gcd(z', s50) :|: s50 >= 0, s50 <= z'', s43 >= 0, s43 <= z', s44 >= 0, s44 <= 2, z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + 0 + 0, z - 1 >= 0 lgth(z) -{ 11 + 6*xs'' }-> s39 :|: s37 >= 0, s37 <= xs'', s38 >= 0, s38 <= 1 + 0 + 0 + s37, s39 >= 0, s39 <= 1 + 0 + 0 + s38, z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 1213 + 982*x11 + 24*x11*xs11 + 12*x11^2 + 988*xs11 + 12*xs11^2 }-> s47 :|: s47 >= 0, s47 <= z, s40 >= 0, s40 <= xs11, s41 >= 0, s41 <= 1 + 0 + 0 + s40, s42 >= 0, s42 <= 2, z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 236 }-> s48 :|: s48 >= 0, s48 <= z, s9 >= 0, s9 <= 2, z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 1449 + 1970*xs + 48*xs^2 }-> s49 :|: s49 >= 0, s49 <= xs', xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] eqList: runtime: O(n^1) [78 + 156*z'], size: O(1) [2] lgth: runtime: O(n^1) [5 + 6*z], size: O(n^1) [z] monus[Ite]: runtime: O(n^2) [958*z'' + 12*z''^2], size: O(n^1) [z'] monus: runtime: O(n^2) [1449 + 1970*z' + 48*z'^2], size: O(n^1) [z] ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: gcd after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z Computed SIZE bound using CoFloCo for: gcd[Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' Computed SIZE bound using CoFloCo for: gcd[False][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 2 + xs }-> 1 + x + s'' :|: s'' >= 0, s'' <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s16 :|: s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s20 :|: s17 >= 0, s17 <= 2, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= 2, s20 >= 0, s20 <= 2, z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s24 :|: s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 159 + 156*y3 + 156*ys3 }-> s28 :|: s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, s27 >= 0, s27 <= 2, s28 >= 0, s28 <= 2, xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 159 + 156*y5 + 156*ys5 }-> s32 :|: s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, s31 >= 0, s31 <= 2, s32 >= 0, s32 <= 2, y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 159 + 156*y7 + 156*ys7 }-> s36 :|: s33 >= 0, s33 <= 2, s34 >= 0, s34 <= 2, s35 >= 0, s35 <= 2, s36 >= 0, s36 <= 2, z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 315 + 156*y' + 156*y1 + 156*ys' + 156*ys1 }-> s8 :|: s2 >= 0, s2 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 158 + 156*x + 156*xs }-> gcd[Ite](s12, 1 + x' + xs', 1 + x + xs) :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 240 + 964*z'' + 12*z''^2 }-> gcd(s51, z'') :|: s51 >= 0, s51 <= z', s45 >= 0, s45 <= z'', s46 >= 0, s46 <= 2, z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 240 + 964*z' + 12*z'^2 }-> gcd(z', s50) :|: s50 >= 0, s50 <= z'', s43 >= 0, s43 <= z', s44 >= 0, s44 <= 2, z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + 0 + 0, z - 1 >= 0 lgth(z) -{ 11 + 6*xs'' }-> s39 :|: s37 >= 0, s37 <= xs'', s38 >= 0, s38 <= 1 + 0 + 0 + s37, s39 >= 0, s39 <= 1 + 0 + 0 + s38, z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 1213 + 982*x11 + 24*x11*xs11 + 12*x11^2 + 988*xs11 + 12*xs11^2 }-> s47 :|: s47 >= 0, s47 <= z, s40 >= 0, s40 <= xs11, s41 >= 0, s41 <= 1 + 0 + 0 + s40, s42 >= 0, s42 <= 2, z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 236 }-> s48 :|: s48 >= 0, s48 <= z, s9 >= 0, s9 <= 2, z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 1449 + 1970*xs + 48*xs^2 }-> s49 :|: s49 >= 0, s49 <= xs', xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] eqList: runtime: O(n^1) [78 + 156*z'], size: O(1) [2] lgth: runtime: O(n^1) [5 + 6*z], size: O(n^1) [z] monus[Ite]: runtime: O(n^2) [958*z'' + 12*z''^2], size: O(n^1) [z'] monus: runtime: O(n^2) [1449 + 1970*z' + 48*z'^2], size: O(n^1) [z] gcd: runtime: ?, size: O(n^1) [z] gcd[Ite]: runtime: ?, size: O(n^1) [z'] gcd[False][Ite]: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: gcd after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 2 + xs }-> 1 + x + s'' :|: s'' >= 0, s'' <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s16 :|: s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, x4 >= 0, ys' >= 0, y' >= 0, xs4 >= 0, xs1 >= 0, z = 1 + (1 + x1 + xs1) + (1 + x4 + xs4) eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s20 :|: s17 >= 0, s17 <= 2, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= 2, s20 >= 0, s20 <= 2, z' = 1 + (1 + y' + ys') + (1 + y2 + ys2), x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0, y2 >= 0, ys2 >= 0 eqList(z, z') -{ 159 + 156*y' + 156*ys' }-> s24 :|: s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, z' = 1 + (1 + y' + ys') + 0, x1 >= 0, z = 1 + (1 + x1 + xs1) + 0, ys' >= 0, y' >= 0, xs1 >= 0 eqList(z, z') -{ 159 + 156*y3 + 156*ys3 }-> s28 :|: s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, s27 >= 0, s27 <= 2, s28 >= 0, s28 <= 2, xs2 >= 0, x5 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x5 + xs5), y3 >= 0, z' = 1 + 0 + (1 + y3 + ys3), ys3 >= 0, xs5 >= 0, x2 >= 0 eqList(z, z') -{ 159 + 156*y5 + 156*ys5 }-> s32 :|: s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, s31 >= 0, s31 <= 2, s32 >= 0, s32 <= 2, y5 >= 0, ys5 >= 0, z = 1 + 0 + (1 + x7 + xs7), x7 >= 0, xs7 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y5 + ys5) eqList(z, z') -{ 159 + 156*y7 + 156*ys7 }-> s36 :|: s33 >= 0, s33 <= 2, s34 >= 0, s34 <= 2, s35 >= 0, s35 <= 2, s36 >= 0, s36 <= 2, z' = 1 + 0 + (1 + y7 + ys7), z = 1 + 0 + (1 + x9 + xs9), y7 >= 0, ys7 >= 0, xs9 >= 0, x9 >= 0 eqList(z, z') -{ 315 + 156*y' + 156*y1 + 156*ys' + 156*ys1 }-> s8 :|: s2 >= 0, s2 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, y1 >= 0, ys1 >= 0, x1 >= 0, ys' >= 0, z = 1 + (1 + x1 + xs1) + (1 + x3 + xs3), y' >= 0, xs1 >= 0, z' = 1 + (1 + y' + ys') + (1 + y1 + ys1), x3 >= 0, xs3 >= 0 eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 3 }-> 1 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), 2 = 2, 1 = 1 eqList(z, z') -{ 3 }-> 0 :|: xs6 >= 0, z = 1 + (1 + x2 + xs2) + (1 + x6 + xs6), xs2 >= 0, z' = 1 + 0 + 0, x6 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z = 1 + (1 + x2 + xs2) + 0, z' = 1 + 0 + (1 + y4 + ys4), ys4 >= 0, y4 >= 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: xs2 >= 0, z' = 1 + 0 + 0, z = 1 + (1 + x2 + xs2) + 0, x2 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: x8 >= 0, z' = 1 + (1 + y'' + ys'') + 0, z = 1 + 0 + (1 + x8 + xs8), y'' >= 0, xs8 >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y6 >= 0, ys6 >= 0, y'' >= 0, ys'' >= 0, z' = 1 + (1 + y'' + ys'') + (1 + y6 + ys6), v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + (1 + y'' + ys'') + 0, y'' >= 0, ys'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x10 >= 0, xs10 >= 0, z = 1 + 0 + (1 + x10 + xs10), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y8 >= 0, ys8 >= 0, z' = 1 + 0 + (1 + y8 + ys8), v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 gcd(z, z') -{ 158 + 156*x + 156*xs }-> gcd[Ite](s12, 1 + x' + xs', 1 + x + xs) :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd[False][Ite](z, z', z'') -{ 240 + 964*z'' + 12*z''^2 }-> gcd(s51, z'') :|: s51 >= 0, s51 <= z', s45 >= 0, s45 <= z'', s46 >= 0, s46 <= 2, z = 2, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 240 + 964*z' + 12*z'^2 }-> gcd(z', s50) :|: s50 >= 0, s50 <= z'', s43 >= 0, s43 <= z', s44 >= 0, s44 <= 2, z = 1, z' >= 0, z'' >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 gcd[Ite](z, z', z'') -{ 2 + xs13 }-> gcd[False][Ite](s', 1 + x''' + xs''', 1 + x13 + xs13) :|: s' >= 0, s' <= 2, xs13 >= 0, x13 >= 0, z = 1, z'' = 1 + x13 + xs13, xs''' >= 0, z' = 1 + x''' + xs''', x''' >= 0 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](2, 1 + x12 + xs12, 0) :|: z'' = 0, z = 1, x12 >= 0, xs12 >= 0, z' = 1 + x12 + xs12 gcd[Ite](z, z', z'') -{ 1 }-> gcd[False][Ite](1, 0, z'') :|: z = 1, z'' >= 0, z' = 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> gcd(z, z') :|: z >= 0, z' >= 0 gt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 2, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 lgth(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + 0 + 0, z - 1 >= 0 lgth(z) -{ 11 + 6*xs'' }-> s39 :|: s37 >= 0, s37 <= xs'', s38 >= 0, s38 <= 1 + 0 + 0 + s37, s39 >= 0, s39 <= 1 + 0 + 0 + s38, z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 monus(z, z') -{ 1213 + 982*x11 + 24*x11*xs11 + 12*x11^2 + 988*xs11 + 12*xs11^2 }-> s47 :|: s47 >= 0, s47 <= z, s40 >= 0, s40 <= xs11, s41 >= 0, s41 <= 1 + 0 + 0 + s40, s42 >= 0, s42 <= 2, z' = 1 + x11 + xs11, z >= 0, xs11 >= 0, x11 >= 0 monus(z, z') -{ 236 }-> s48 :|: s48 >= 0, s48 <= z, s9 >= 0, s9 <= 2, z >= 0, z' = 0 monus[Ite](z, z', z'') -{ 1449 + 1970*xs + 48*xs^2 }-> s49 :|: s49 >= 0, s49 <= xs', xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0 monus[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 Function symbols to be analyzed: {gcd,gcd[Ite],gcd[False][Ite]}, {goal} Previous analysis results are: gt0: runtime: O(n^1) [1 + z'], size: O(1) [2] and: runtime: O(1) [0], size: O(1) [2] @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] eqList: runtime: O(n^1) [78 + 156*z'], size: O(1) [2] lgth: runtime: O(n^1) [5 + 6*z], size: O(n^1) [z] monus[Ite]: runtime: O(n^2) [958*z'' + 12*z''^2], size: O(n^1) [z'] monus: runtime: O(n^2) [1449 + 1970*z' + 48*z'^2], size: O(n^1) [z] gcd: runtime: INF, size: O(n^1) [z] gcd[Ite]: runtime: ?, size: O(n^1) [z'] gcd[False][Ite]: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (65) 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: and(v0, v1) -> null_and [0] monus[Ite](v0, v1, v2) -> null_monus[Ite] [0] gcd[Ite](v0, v1, v2) -> null_gcd[Ite] [0] gcd[False][Ite](v0, v1, v2) -> null_gcd[False][Ite] [0] @(v0, v1) -> null_@ [0] gt0(v0, v1) -> null_gt0 [0] gcd(v0, v1) -> null_gcd [0] lgth(v0) -> null_lgth [0] eqList(v0, v1) -> null_eqList [0] And the following fresh constants: null_and, null_monus[Ite], null_gcd[Ite], null_gcd[False][Ite], null_@, null_gt0, null_gcd, null_lgth, null_eqList ---------------------------------------- (66) 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: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) [1] @(Nil, ys) -> ys [1] gt0(Cons(x, xs), Nil) -> True [1] gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) [1] gcd(Nil, Nil) -> Nil [1] gcd(Nil, Cons(x, xs)) -> Nil [1] gcd(Cons(x, xs), Nil) -> Nil [1] gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs)) [1] lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs)) [1] eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys)) [1] eqList(Cons(x, xs), Nil) -> False [1] eqList(Nil, Cons(y, ys)) -> False [1] eqList(Nil, Nil) -> True [1] lgth(Nil) -> Nil [1] gt0(Nil, y) -> False [1] monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y) [1] goal(x, y) -> gcd(x, y) [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) [0] monus[Ite](True, Cons(x, xs), y) -> xs [0] gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y) [0] gcd[Ite](True, x, y) -> x [0] gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x)) [0] gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y) [0] and(v0, v1) -> null_and [0] monus[Ite](v0, v1, v2) -> null_monus[Ite] [0] gcd[Ite](v0, v1, v2) -> null_gcd[Ite] [0] gcd[False][Ite](v0, v1, v2) -> null_gcd[False][Ite] [0] @(v0, v1) -> null_@ [0] gt0(v0, v1) -> null_gt0 [0] gcd(v0, v1) -> null_gcd [0] lgth(v0) -> null_lgth [0] eqList(v0, v1) -> null_eqList [0] The TRS has the following type information: @ :: Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth Cons :: Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth Nil :: Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth gt0 :: Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> True:False:null_and:null_gt0:null_eqList True :: True:False:null_and:null_gt0:null_eqList gcd :: Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth gcd[Ite] :: True:False:null_and:null_gt0:null_eqList -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth eqList :: Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> True:False:null_and:null_gt0:null_eqList lgth :: Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth and :: True:False:null_and:null_gt0:null_eqList -> True:False:null_and:null_gt0:null_eqList -> True:False:null_and:null_gt0:null_eqList False :: True:False:null_and:null_gt0:null_eqList monus :: Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth monus[Ite] :: True:False:null_and:null_gt0:null_eqList -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth goal :: Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth gcd[False][Ite] :: True:False:null_and:null_gt0:null_eqList -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth -> Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth null_and :: True:False:null_and:null_gt0:null_eqList null_monus[Ite] :: Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth null_gcd[Ite] :: Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth null_gcd[False][Ite] :: Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth null_@ :: Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth null_gt0 :: True:False:null_and:null_gt0:null_eqList null_gcd :: Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth null_lgth :: Cons:Nil:null_monus[Ite]:null_gcd[Ite]:null_gcd[False][Ite]:null_@:null_gcd:null_lgth null_eqList :: True:False:null_and:null_gt0:null_eqList Rewrite Strategy: INNERMOST ---------------------------------------- (67) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: Nil => 0 True => 2 False => 1 null_and => 0 null_monus[Ite] => 0 null_gcd[Ite] => 0 null_gcd[False][Ite] => 0 null_@ => 0 null_gt0 => 0 null_gcd => 0 null_lgth => 0 null_eqList => 0 ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0 @(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 @(z, z') -{ 1 }-> 1 + x + @(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 eqList(z, z') -{ 1 }-> and(eqList(x, y), eqList(xs, ys)) :|: z = 1 + x + xs, xs >= 0, ys >= 0, x >= 0, y >= 0, z' = 1 + y + ys eqList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 eqList(z, z') -{ 1 }-> 1 :|: ys >= 0, y >= 0, z = 0, z' = 1 + y + ys eqList(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 gcd(z, z') -{ 1 }-> gcd[Ite](eqList(1 + x' + xs', 1 + x + xs), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gcd(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 gcd(z, z') -{ 1 }-> 0 :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 gcd(z, z') -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gcd(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 gcd[False][Ite](z, z', z'') -{ 0 }-> gcd(x, monus(y, x)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> gcd(monus(x, y), y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 gcd[False][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 gcd[Ite](z, z', z'') -{ 0 }-> x :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 gcd[Ite](z, z', z'') -{ 0 }-> gcd[False][Ite](gt0(x, y), x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 gcd[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 goal(z, z') -{ 1 }-> gcd(x, y) :|: x >= 0, y >= 0, z = x, z' = y gt0(z, z') -{ 1 }-> gt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' gt0(z, z') -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 gt0(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y gt0(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 lgth(z) -{ 1 }-> @(1 + 0 + 0, lgth(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 lgth(z) -{ 1 }-> 0 :|: z = 0 lgth(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 monus(z, z') -{ 1 }-> monus[Ite](eqList(lgth(y), 1 + 0 + 0), x, y) :|: x >= 0, y >= 0, z = x, z' = y monus[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' = 1 + x + xs, z'' = y, x >= 0, y >= 0 monus[Ite](z, z', z'') -{ 0 }-> monus(xs', xs) :|: xs >= 0, z = 1, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x' + xs', z'' = 1 + x + xs monus[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (69) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 gcd[Ite](False, z0, z1) -> gcd[False][Ite](gt0(z0, z1), z0, z1) gcd[Ite](True, z0, z1) -> z0 gcd[False][Ite](False, z0, z1) -> gcd(z0, monus(z1, z0)) gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False gcd(Nil, Nil) -> Nil gcd(Nil, Cons(z0, z1)) -> Nil gcd(Cons(z0, z1), Nil) -> Nil gcd(Cons(z0, z1), Cons(z2, z3)) -> gcd[Ite](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)) lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Cons(z0, z1)) -> False eqList(Nil, Nil) -> True monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) goal(z0, z1) -> gcd(z0, z1) Tuples: AND(False, False) -> c AND(True, False) -> c1 AND(False, True) -> c2 AND(True, True) -> c3 MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) MONUS[ITE](True, Cons(z0, z1), z2) -> c5 GCD[ITE](False, z0, z1) -> c6(GCD[FALSE][ITE](gt0(z0, z1), z0, z1), GT0(z0, z1)) GCD[ITE](True, z0, z1) -> c7 GCD[FALSE][ITE](False, z0, z1) -> c8(GCD(z0, monus(z1, z0)), MONUS(z1, z0)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus(z0, z1), z1), MONUS(z0, z1)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) @'(Nil, z0) -> c11 GT0(Cons(z0, z1), Nil) -> c12 GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GT0(Nil, z0) -> c14 GCD(Nil, Nil) -> c15 GCD(Nil, Cons(z0, z1)) -> c16 GCD(Cons(z0, z1), Nil) -> c17 GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) LGTH(Nil) -> c20 EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z1, z3)) EQLIST(Cons(z0, z1), Nil) -> c23 EQLIST(Nil, Cons(z0, z1)) -> c24 EQLIST(Nil, Nil) -> c25 MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) GOAL(z0, z1) -> c27(GCD(z0, z1)) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) @'(Nil, z0) -> c11 GT0(Cons(z0, z1), Nil) -> c12 GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GT0(Nil, z0) -> c14 GCD(Nil, Nil) -> c15 GCD(Nil, Cons(z0, z1)) -> c16 GCD(Cons(z0, z1), Nil) -> c17 GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) LGTH(Nil) -> c20 EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z1, z3)) EQLIST(Cons(z0, z1), Nil) -> c23 EQLIST(Nil, Cons(z0, z1)) -> c24 EQLIST(Nil, Nil) -> c25 MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) GOAL(z0, z1) -> c27(GCD(z0, z1)) K tuples:none Defined Rule Symbols: @_2, gt0_2, gcd_2, lgth_1, eqList_2, monus_2, goal_2, and_2, monus[Ite]_3, gcd[Ite]_3, gcd[False][Ite]_3 Defined Pair Symbols: AND_2, MONUS[ITE]_3, GCD[ITE]_3, GCD[FALSE][ITE]_3, @'_2, GT0_2, GCD_2, LGTH_1, EQLIST_2, MONUS_2, GOAL_2 Compound Symbols: c, c1, c2, c3, c4_1, c5, c6_2, c7, c8_2, c9_2, c10_1, c11, c12, c13_1, c14, c15, c16, c17, c18_2, c19_2, c20, c21_2, c22_2, c23, c24, c25, c26_3, c27_1 ---------------------------------------- (71) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: GOAL(z0, z1) -> c27(GCD(z0, z1)) Removed 11 trailing nodes: EQLIST(Nil, Cons(z0, z1)) -> c24 AND(True, True) -> c3 AND(True, False) -> c1 AND(False, False) -> c MONUS[ITE](True, Cons(z0, z1), z2) -> c5 AND(False, True) -> c2 LGTH(Nil) -> c20 @'(Nil, z0) -> c11 GCD[ITE](True, z0, z1) -> c7 EQLIST(Nil, Nil) -> c25 EQLIST(Cons(z0, z1), Nil) -> c23 ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 gcd[Ite](False, z0, z1) -> gcd[False][Ite](gt0(z0, z1), z0, z1) gcd[Ite](True, z0, z1) -> z0 gcd[False][Ite](False, z0, z1) -> gcd(z0, monus(z1, z0)) gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False gcd(Nil, Nil) -> Nil gcd(Nil, Cons(z0, z1)) -> Nil gcd(Cons(z0, z1), Nil) -> Nil gcd(Cons(z0, z1), Cons(z2, z3)) -> gcd[Ite](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)) lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Cons(z0, z1)) -> False eqList(Nil, Nil) -> True monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) goal(z0, z1) -> gcd(z0, z1) Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) GCD[ITE](False, z0, z1) -> c6(GCD[FALSE][ITE](gt0(z0, z1), z0, z1), GT0(z0, z1)) GCD[FALSE][ITE](False, z0, z1) -> c8(GCD(z0, monus(z1, z0)), MONUS(z1, z0)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus(z0, z1), z1), MONUS(z0, z1)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Nil) -> c12 GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GT0(Nil, z0) -> c14 GCD(Nil, Nil) -> c15 GCD(Nil, Cons(z0, z1)) -> c16 GCD(Cons(z0, z1), Nil) -> c17 GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z1, z3)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Nil) -> c12 GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GT0(Nil, z0) -> c14 GCD(Nil, Nil) -> c15 GCD(Nil, Cons(z0, z1)) -> c16 GCD(Cons(z0, z1), Nil) -> c17 GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z1, z3)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) K tuples:none Defined Rule Symbols: @_2, gt0_2, gcd_2, lgth_1, eqList_2, monus_2, goal_2, and_2, monus[Ite]_3, gcd[Ite]_3, gcd[False][Ite]_3 Defined Pair Symbols: MONUS[ITE]_3, GCD[ITE]_3, GCD[FALSE][ITE]_3, @'_2, GT0_2, GCD_2, LGTH_1, EQLIST_2, MONUS_2 Compound Symbols: c4_1, c6_2, c8_2, c9_2, c10_1, c12, c13_1, c14, c15, c16, c17, c18_2, c19_2, c21_2, c22_2, c26_3 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 gcd[Ite](False, z0, z1) -> gcd[False][Ite](gt0(z0, z1), z0, z1) gcd[Ite](True, z0, z1) -> z0 gcd[False][Ite](False, z0, z1) -> gcd(z0, monus(z1, z0)) gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False gcd(Nil, Nil) -> Nil gcd(Nil, Cons(z0, z1)) -> Nil gcd(Cons(z0, z1), Nil) -> Nil gcd(Cons(z0, z1), Cons(z2, z3)) -> gcd[Ite](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)) lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Cons(z0, z1)) -> False eqList(Nil, Nil) -> True monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) goal(z0, z1) -> gcd(z0, z1) Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) GCD[ITE](False, z0, z1) -> c6(GCD[FALSE][ITE](gt0(z0, z1), z0, z1), GT0(z0, z1)) GCD[FALSE][ITE](False, z0, z1) -> c8(GCD(z0, monus(z1, z0)), MONUS(z1, z0)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus(z0, z1), z1), MONUS(z0, z1)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Nil) -> c12 GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GT0(Nil, z0) -> c14 GCD(Nil, Nil) -> c15 GCD(Nil, Cons(z0, z1)) -> c16 GCD(Cons(z0, z1), Nil) -> c17 GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Nil) -> c12 GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GT0(Nil, z0) -> c14 GCD(Nil, Nil) -> c15 GCD(Nil, Cons(z0, z1)) -> c16 GCD(Cons(z0, z1), Nil) -> c17 GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) K tuples:none Defined Rule Symbols: @_2, gt0_2, gcd_2, lgth_1, eqList_2, monus_2, goal_2, and_2, monus[Ite]_3, gcd[Ite]_3, gcd[False][Ite]_3 Defined Pair Symbols: MONUS[ITE]_3, GCD[ITE]_3, GCD[FALSE][ITE]_3, @'_2, GT0_2, GCD_2, LGTH_1, MONUS_2, EQLIST_2 Compound Symbols: c4_1, c6_2, c8_2, c9_2, c10_1, c12, c13_1, c14, c15, c16, c17, c18_2, c19_2, c26_3, c21_1, c22_1 ---------------------------------------- (75) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: gcd[Ite](False, z0, z1) -> gcd[False][Ite](gt0(z0, z1), z0, z1) gcd[Ite](True, z0, z1) -> z0 gcd[False][Ite](False, z0, z1) -> gcd(z0, monus(z1, z0)) gcd[False][Ite](True, z0, z1) -> gcd(monus(z0, z1), z1) gcd(Nil, Nil) -> Nil gcd(Nil, Cons(z0, z1)) -> Nil gcd(Cons(z0, z1), Nil) -> Nil gcd(Cons(z0, z1), Cons(z2, z3)) -> gcd[Ite](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)) goal(z0, z1) -> gcd(z0, z1) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) GCD[ITE](False, z0, z1) -> c6(GCD[FALSE][ITE](gt0(z0, z1), z0, z1), GT0(z0, z1)) GCD[FALSE][ITE](False, z0, z1) -> c8(GCD(z0, monus(z1, z0)), MONUS(z1, z0)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus(z0, z1), z1), MONUS(z0, z1)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Nil) -> c12 GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GT0(Nil, z0) -> c14 GCD(Nil, Nil) -> c15 GCD(Nil, Cons(z0, z1)) -> c16 GCD(Cons(z0, z1), Nil) -> c17 GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Nil) -> c12 GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GT0(Nil, z0) -> c14 GCD(Nil, Nil) -> c15 GCD(Nil, Cons(z0, z1)) -> c16 GCD(Cons(z0, z1), Nil) -> c17 GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, GCD[ITE]_3, GCD[FALSE][ITE]_3, @'_2, GT0_2, GCD_2, LGTH_1, MONUS_2, EQLIST_2 Compound Symbols: c4_1, c6_2, c8_2, c9_2, c10_1, c12, c13_1, c14, c15, c16, c17, c18_2, c19_2, c26_3, c21_1, c22_1 ---------------------------------------- (77) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. GCD(Nil, Nil) -> c15 GCD(Nil, Cons(z0, z1)) -> c16 GCD(Cons(z0, z1), Nil) -> c17 We considered the (Usable) Rules:none And the Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) GCD[ITE](False, z0, z1) -> c6(GCD[FALSE][ITE](gt0(z0, z1), z0, z1), GT0(z0, z1)) GCD[FALSE][ITE](False, z0, z1) -> c8(GCD(z0, monus(z1, z0)), MONUS(z1, z0)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus(z0, z1), z1), MONUS(z0, z1)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Nil) -> c12 GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GT0(Nil, z0) -> c14 GCD(Nil, Nil) -> c15 GCD(Nil, Cons(z0, z1)) -> c16 GCD(Cons(z0, z1), Nil) -> c17 GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) The order we found is given by the following interpretation: Polynomial interpretation : POL(@(x_1, x_2)) = [1] + x_1 + x_2 POL(@'(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = 0 POL(EQLIST(x_1, x_2)) = 0 POL(False) = 0 POL(GCD(x_1, x_2)) = [1] POL(GCD[FALSE][ITE](x_1, x_2, x_3)) = [1] POL(GCD[ITE](x_1, x_2, x_3)) = [1] POL(GT0(x_1, x_2)) = 0 POL(LGTH(x_1)) = 0 POL(MONUS(x_1, x_2)) = 0 POL(MONUS[ITE](x_1, x_2, x_3)) = 0 POL(Nil) = 0 POL(True) = 0 POL(and(x_1, x_2)) = [1] POL(c10(x_1)) = x_1 POL(c12) = 0 POL(c13(x_1)) = x_1 POL(c14) = 0 POL(c15) = 0 POL(c16) = 0 POL(c17) = 0 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2)) = x_1 + x_2 POL(eqList(x_1, x_2)) = 0 POL(gt0(x_1, x_2)) = [1] POL(lgth(x_1)) = x_1 POL(monus(x_1, x_2)) = [1] + x_1 + x_2 POL(monus[Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) GCD[ITE](False, z0, z1) -> c6(GCD[FALSE][ITE](gt0(z0, z1), z0, z1), GT0(z0, z1)) GCD[FALSE][ITE](False, z0, z1) -> c8(GCD(z0, monus(z1, z0)), MONUS(z1, z0)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus(z0, z1), z1), MONUS(z0, z1)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Nil) -> c12 GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GT0(Nil, z0) -> c14 GCD(Nil, Nil) -> c15 GCD(Nil, Cons(z0, z1)) -> c16 GCD(Cons(z0, z1), Nil) -> c17 GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Nil) -> c12 GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GT0(Nil, z0) -> c14 GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) K tuples: GCD(Nil, Nil) -> c15 GCD(Nil, Cons(z0, z1)) -> c16 GCD(Cons(z0, z1), Nil) -> c17 Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, GCD[ITE]_3, GCD[FALSE][ITE]_3, @'_2, GT0_2, GCD_2, LGTH_1, MONUS_2, EQLIST_2 Compound Symbols: c4_1, c6_2, c8_2, c9_2, c10_1, c12, c13_1, c14, c15, c16, c17, c18_2, c19_2, c26_3, c21_1, c22_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD[ITE](False, z0, z1) -> c6(GCD[FALSE][ITE](gt0(z0, z1), z0, z1), GT0(z0, z1)) by GCD[ITE](False, Cons(z0, z1), Nil) -> c6(GCD[FALSE][ITE](True, Cons(z0, z1), Nil), GT0(Cons(z0, z1), Nil)) GCD[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(z0, z1), Cons(z2, z3)), GT0(Cons(z0, z1), Cons(z2, z3))) GCD[ITE](False, Nil, z0) -> c6(GCD[FALSE][ITE](False, Nil, z0), GT0(Nil, z0)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) GCD[FALSE][ITE](False, z0, z1) -> c8(GCD(z0, monus(z1, z0)), MONUS(z1, z0)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus(z0, z1), z1), MONUS(z0, z1)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Nil) -> c12 GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GT0(Nil, z0) -> c14 GCD(Nil, Nil) -> c15 GCD(Nil, Cons(z0, z1)) -> c16 GCD(Cons(z0, z1), Nil) -> c17 GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(z0, z1), Nil) -> c6(GCD[FALSE][ITE](True, Cons(z0, z1), Nil), GT0(Cons(z0, z1), Nil)) GCD[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(z0, z1), Cons(z2, z3)), GT0(Cons(z0, z1), Cons(z2, z3))) GCD[ITE](False, Nil, z0) -> c6(GCD[FALSE][ITE](False, Nil, z0), GT0(Nil, z0)) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Nil) -> c12 GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GT0(Nil, z0) -> c14 GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) K tuples: GCD(Nil, Nil) -> c15 GCD(Nil, Cons(z0, z1)) -> c16 GCD(Cons(z0, z1), Nil) -> c17 Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, GCD[FALSE][ITE]_3, @'_2, GT0_2, GCD_2, LGTH_1, MONUS_2, EQLIST_2, GCD[ITE]_3 Compound Symbols: c4_1, c8_2, c9_2, c10_1, c12, c13_1, c14, c15, c16, c17, c18_2, c19_2, c26_3, c21_1, c22_1, c6_2 ---------------------------------------- (81) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: GCD[ITE](False, Cons(z0, z1), Nil) -> c6(GCD[FALSE][ITE](True, Cons(z0, z1), Nil), GT0(Cons(z0, z1), Nil)) GCD[ITE](False, Nil, z0) -> c6(GCD[FALSE][ITE](False, Nil, z0), GT0(Nil, z0)) Removed 3 trailing nodes: GCD(Cons(z0, z1), Nil) -> c17 GCD(Nil, Cons(z0, z1)) -> c16 GCD(Nil, Nil) -> c15 ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) GCD[FALSE][ITE](False, z0, z1) -> c8(GCD(z0, monus(z1, z0)), MONUS(z1, z0)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus(z0, z1), z1), MONUS(z0, z1)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Nil) -> c12 GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GT0(Nil, z0) -> c14 GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(z0, z1), Cons(z2, z3)), GT0(Cons(z0, z1), Cons(z2, z3))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Nil) -> c12 GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GT0(Nil, z0) -> c14 GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, GCD[FALSE][ITE]_3, @'_2, GT0_2, GCD_2, LGTH_1, MONUS_2, EQLIST_2, GCD[ITE]_3 Compound Symbols: c4_1, c8_2, c9_2, c10_1, c12, c13_1, c14, c18_2, c19_2, c26_3, c21_1, c22_1, c6_2 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: GT0(Cons(z0, z1), Nil) -> c12 GT0(Nil, z0) -> c14 ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) GCD[FALSE][ITE](False, z0, z1) -> c8(GCD(z0, monus(z1, z0)), MONUS(z1, z0)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus(z0, z1), z1), MONUS(z0, z1)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(z0, z1), Cons(z2, z3)), GT0(Cons(z0, z1), Cons(z2, z3))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, GCD[FALSE][ITE]_3, @'_2, GT0_2, GCD_2, LGTH_1, MONUS_2, EQLIST_2, GCD[ITE]_3 Compound Symbols: c4_1, c8_2, c9_2, c10_1, c13_1, c18_2, c19_2, c26_3, c21_1, c22_1, c6_2 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD[FALSE][ITE](False, z0, z1) -> c8(GCD(z0, monus(z1, z0)), MONUS(z1, z0)) by GCD[FALSE][ITE](False, z1, z0) -> c8(GCD(z1, monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1)), MONUS(z0, z1)) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus(z0, z1), z1), MONUS(z0, z1)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(z0, z1), Cons(z2, z3)), GT0(Cons(z0, z1), Cons(z2, z3))) GCD[FALSE][ITE](False, z1, z0) -> c8(GCD(z1, monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1)), MONUS(z0, z1)) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, GCD[FALSE][ITE]_3, @'_2, GT0_2, GCD_2, LGTH_1, MONUS_2, EQLIST_2, GCD[ITE]_3 Compound Symbols: c4_1, c9_2, c10_1, c13_1, c18_2, c19_2, c26_3, c21_1, c22_1, c6_2, c8_2 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus(z0, z1), z1), MONUS(z0, z1)) by GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), z1), MONUS(z0, z1)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(z0, z1), Cons(z2, z3)), GT0(Cons(z0, z1), Cons(z2, z3))) GCD[FALSE][ITE](False, z1, z0) -> c8(GCD(z1, monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1)), MONUS(z0, z1)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), z1), MONUS(z0, z1)) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, GCD_2, LGTH_1, MONUS_2, EQLIST_2, GCD[ITE]_3, GCD[FALSE][ITE]_3 Compound Symbols: c4_1, c10_1, c13_1, c18_2, c19_2, c26_3, c21_1, c22_1, c6_2, c8_2, c9_2 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) by GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(z0, z1), Cons(z2, z3)), GT0(Cons(z0, z1), Cons(z2, z3))) GCD[FALSE][ITE](False, z1, z0) -> c8(GCD(z1, monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1)), MONUS(z0, z1)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), z1), MONUS(z0, z1)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, MONUS_2, EQLIST_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, GCD_2 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c26_3, c21_1, c22_1, c6_2, c8_2, c9_2, c18_2 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1)) by MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(MONUS[ITE](eqList(Nil, Cons(Nil, Nil)), x0, Nil), EQLIST(lgth(Nil), Cons(Nil, Nil)), LGTH(Nil)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(z0, z1), Cons(z2, z3)), GT0(Cons(z0, z1), Cons(z2, z3))) GCD[FALSE][ITE](False, z1, z0) -> c8(GCD(z1, monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1)), MONUS(z0, z1)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), z1), MONUS(z0, z1)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(MONUS[ITE](eqList(Nil, Cons(Nil, Nil)), x0, Nil), EQLIST(lgth(Nil), Cons(Nil, Nil)), LGTH(Nil)) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(MONUS[ITE](eqList(Nil, Cons(Nil, Nil)), x0, Nil), EQLIST(lgth(Nil), Cons(Nil, Nil)), LGTH(Nil)) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, GCD_2, MONUS_2 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c6_2, c8_2, c9_2, c18_2, c26_3 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(z0, z1), Cons(z2, z3)), GT0(Cons(z0, z1), Cons(z2, z3))) GCD[FALSE][ITE](False, z1, z0) -> c8(GCD(z1, monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1)), MONUS(z0, z1)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), z1), MONUS(z0, z1)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, GCD_2, MONUS_2 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c6_2, c8_2, c9_2, c18_2, c26_3, c26_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(z0, z1), Cons(z2, z3)), GT0(Cons(z0, z1), Cons(z2, z3))) by GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[FALSE][ITE](False, z1, z0) -> c8(GCD(z1, monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1)), MONUS(z0, z1)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), z1), MONUS(z0, z1)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, GCD[FALSE][ITE]_3, GCD_2, MONUS_2, GCD[ITE]_3 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c8_2, c9_2, c18_2, c26_3, c26_1, c6_2, c6_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD[FALSE][ITE](False, z1, z0) -> c8(GCD(z1, monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1)), MONUS(z0, z1)) by GCD[FALSE][ITE](False, Cons(z0, z1), x1) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x1, Cons(z0, z1))), MONUS(x1, Cons(z0, z1))) GCD[FALSE][ITE](False, Nil, x1) -> c8(GCD(Nil, monus[Ite](eqList(Nil, Cons(Nil, Nil)), x1, Nil)), MONUS(x1, Nil)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), z1), MONUS(z0, z1)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](False, Cons(z0, z1), x1) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x1, Cons(z0, z1))), MONUS(x1, Cons(z0, z1))) GCD[FALSE][ITE](False, Nil, x1) -> c8(GCD(Nil, monus[Ite](eqList(Nil, Cons(Nil, Nil)), x1, Nil)), MONUS(x1, Nil)) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, GCD[FALSE][ITE]_3, GCD_2, MONUS_2, GCD[ITE]_3 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c9_2, c18_2, c26_3, c26_1, c6_2, c6_1, c8_2 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), z1), MONUS(z0, z1)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](False, Cons(z0, z1), x1) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x1, Cons(z0, z1))), MONUS(x1, Cons(z0, z1))) GCD[FALSE][ITE](False, Nil, x1) -> c8(MONUS(x1, Nil)) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, GCD[FALSE][ITE]_3, GCD_2, MONUS_2, GCD[ITE]_3 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c9_2, c18_2, c26_3, c26_1, c6_2, c6_1, c8_2, c8_1 ---------------------------------------- (101) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: GCD[FALSE][ITE](False, Nil, x1) -> c8(MONUS(x1, Nil)) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), z1), MONUS(z0, z1)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](False, Cons(z0, z1), x1) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x1, Cons(z0, z1))), MONUS(x1, Cons(z0, z1))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, GCD[FALSE][ITE]_3, GCD_2, MONUS_2, GCD[ITE]_3 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c9_2, c18_2, c26_3, c26_1, c6_2, c6_1, c8_2 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), z1), MONUS(z0, z1)) by GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) GCD[FALSE][ITE](True, x0, Nil) -> c9(GCD(monus[Ite](eqList(Nil, Cons(Nil, Nil)), x0, Nil), Nil), MONUS(x0, Nil)) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](False, Cons(z0, z1), x1) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x1, Cons(z0, z1))), MONUS(x1, Cons(z0, z1))) GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) GCD[FALSE][ITE](True, x0, Nil) -> c9(GCD(monus[Ite](eqList(Nil, Cons(Nil, Nil)), x0, Nil), Nil), MONUS(x0, Nil)) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, GCD_2, MONUS_2, GCD[ITE]_3, GCD[FALSE][ITE]_3 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c18_2, c26_3, c26_1, c6_2, c6_1, c8_2, c9_2 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](False, Cons(z0, z1), x1) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x1, Cons(z0, z1))), MONUS(x1, Cons(z0, z1))) GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) GCD[FALSE][ITE](True, x0, Nil) -> c9(MONUS(x0, Nil)) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, GCD_2, MONUS_2, GCD[ITE]_3, GCD[FALSE][ITE]_3 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c18_2, c26_3, c26_1, c6_2, c6_1, c8_2, c9_2, c9_1 ---------------------------------------- (107) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: GCD[FALSE][ITE](True, x0, Nil) -> c9(MONUS(x0, Nil)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](False, Cons(z0, z1), x1) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x1, Cons(z0, z1))), MONUS(x1, Cons(z0, z1))) GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, GCD_2, MONUS_2, GCD[ITE]_3, GCD[FALSE][ITE]_3 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c18_2, c26_3, c26_1, c6_2, c6_1, c8_2, c9_2 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](and(eqList(z0, z2), eqList(z1, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3))) by GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](False, Cons(z0, z1), x1) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x1, Cons(z0, z1))), MONUS(x1, Cons(z0, z1))) GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) K tuples:none Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, MONUS_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, GCD_2 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c26_3, c26_1, c6_2, c6_1, c8_2, c9_2, c18_2, c18_1 ---------------------------------------- (111) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) We considered the (Usable) Rules: gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False gt0(Cons(z0, z1), Nil) -> True And the Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](False, Cons(z0, z1), x1) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x1, Cons(z0, z1))), MONUS(x1, Cons(z0, z1))) GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) The order we found is given by the following interpretation: Polynomial interpretation : POL(@(x_1, x_2)) = [1] + x_1 POL(@'(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [1] POL(EQLIST(x_1, x_2)) = 0 POL(False) = [1] POL(GCD(x_1, x_2)) = [1] POL(GCD[FALSE][ITE](x_1, x_2, x_3)) = x_1 POL(GCD[ITE](x_1, x_2, x_3)) = x_3 POL(GT0(x_1, x_2)) = 0 POL(LGTH(x_1)) = 0 POL(MONUS(x_1, x_2)) = 0 POL(MONUS[ITE](x_1, x_2, x_3)) = 0 POL(Nil) = 0 POL(True) = [1] POL(and(x_1, x_2)) = 0 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2)) = x_1 + x_2 POL(eqList(x_1, x_2)) = 0 POL(gt0(x_1, x_2)) = [1] POL(lgth(x_1)) = 0 POL(monus(x_1, x_2)) = [1] + x_1 + x_2 POL(monus[Ite](x_1, x_2, x_3)) = [1] + x_3 ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](False, Cons(z0, z1), x1) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x1, Cons(z0, z1))), MONUS(x1, Cons(z0, z1))) GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, MONUS_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, GCD_2 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c26_3, c26_1, c6_2, c6_1, c8_2, c9_2, c18_2, c18_1 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MONUS(x0, Cons(z0, z1)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), EQLIST(lgth(Cons(z0, z1)), Cons(Nil, Nil)), LGTH(Cons(z0, z1))) by MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, Nil)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil)), EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil)), LGTH(Cons(x1, Nil))) MONUS(x0, Cons(x1, x2)) -> c26(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](False, Cons(z0, z1), x1) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x1, Cons(z0, z1))), MONUS(x1, Cons(z0, z1))) GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, Nil)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil)), EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil)), LGTH(Cons(x1, Nil))) MONUS(x0, Cons(x1, x2)) -> c26(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, Nil)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil)), EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil)), LGTH(Cons(x1, Nil))) MONUS(x0, Cons(x1, x2)) -> c26(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, MONUS_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, GCD_2 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c26_1, c6_2, c6_1, c8_2, c9_2, c18_2, c18_1, c26_3, c26_2 ---------------------------------------- (115) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](False, Cons(z0, z1), x1) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x1, Cons(z0, z1))), MONUS(x1, Cons(z0, z1))) GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, Nil)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil)), EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil)), LGTH(Cons(x1, Nil))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, Nil)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil)), EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil)), LGTH(Cons(x1, Nil))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, MONUS_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, GCD_2 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c26_1, c6_2, c6_1, c8_2, c9_2, c18_2, c18_1, c26_3, c_1 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MONUS(x0, Nil) -> c26(EQLIST(lgth(Nil), Cons(Nil, Nil))) by MONUS(x0, Nil) -> c26(EQLIST(Nil, Cons(Nil, Nil))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](False, Cons(z0, z1), x1) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x1, Cons(z0, z1))), MONUS(x1, Cons(z0, z1))) GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, Nil)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil)), EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil)), LGTH(Cons(x1, Nil))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26(EQLIST(Nil, Cons(Nil, Nil))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, Nil)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil)), EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil)), LGTH(Cons(x1, Nil))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26(EQLIST(Nil, Cons(Nil, Nil))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, GCD_2, MONUS_2 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c6_2, c6_1, c8_2, c9_2, c18_2, c18_1, c26_3, c_1, c26_1 ---------------------------------------- (119) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](False, Cons(z0, z1), x1) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x1, Cons(z0, z1))), MONUS(x1, Cons(z0, z1))) GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, Nil)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil)), EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil)), LGTH(Cons(x1, Nil))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, Nil)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil)), EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil)), LGTH(Cons(x1, Nil))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, GCD_2, MONUS_2 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c6_2, c6_1, c8_2, c9_2, c18_2, c18_1, c26_3, c_1, c26 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GCD[FALSE][ITE](False, Cons(z0, z1), x1) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x1, Cons(z0, z1))), MONUS(x1, Cons(z0, z1))) by GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, Nil)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil)), EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil)), LGTH(Cons(x1, Nil))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, Nil)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil)), EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil)), LGTH(Cons(x1, Nil))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: MONUS[ITE]_3, @'_2, GT0_2, LGTH_1, EQLIST_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, GCD_2, MONUS_2 Compound Symbols: c4_1, c10_1, c13_1, c19_2, c21_1, c22_1, c6_2, c6_1, c9_2, c18_2, c18_1, c26_3, c_1, c26, c8_2 ---------------------------------------- (123) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3)) by MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, Nil)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil)), EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil)), LGTH(Cons(x1, Nil))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, Nil)) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil)), EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil)), LGTH(Cons(x1, Nil))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: @'_2, GT0_2, LGTH_1, EQLIST_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3 Compound Symbols: c10_1, c13_1, c19_2, c21_1, c22_1, c6_2, c6_1, c9_2, c18_2, c18_1, c26_3, c_1, c26, c8_2, c4_1 ---------------------------------------- (125) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: @'_2, GT0_2, LGTH_1, EQLIST_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3 Compound Symbols: c10_1, c13_1, c19_2, c21_1, c22_1, c6_2, c6_1, c9_2, c18_2, c18_1, c26_3, c_1, c26, c8_2, c4_1, c1_1 ---------------------------------------- (127) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace GCD[ITE](False, Cons(x0, Nil), Cons(x2, z0)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x2, z0)), GT0(Cons(x0, Nil), Cons(x2, z0))) by GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))), GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)), GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)), GT0(Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)), GT0(Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)), GT0(Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1)), GT0(Cons(Nil, Nil), Cons(Nil, x1))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))), GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)), GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)), GT0(Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)), GT0(Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)), GT0(Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1)), GT0(Cons(Nil, Nil), Cons(Nil, x1))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: @'_2, GT0_2, LGTH_1, EQLIST_2, GCD[ITE]_3, GCD[FALSE][ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3 Compound Symbols: c10_1, c13_1, c19_2, c21_1, c22_1, c6_2, c6_1, c9_2, c18_2, c18_1, c26_3, c_1, c26, c8_2, c4_1, c1_1 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GCD[FALSE][ITE](True, x0, Cons(z0, z1)) -> c9(GCD(monus[Ite](eqList(@(Cons(Nil, Nil), lgth(z1)), Cons(Nil, Nil)), x0, Cons(z0, z1)), Cons(z0, z1)), MONUS(x0, Cons(z0, z1))) by GCD[FALSE][ITE](True, z0, Cons(z1, z2)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), Cons(z1, z2)), MONUS(z0, Cons(z1, z2))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))), GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)), GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)), GT0(Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)), GT0(Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)), GT0(Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1)), GT0(Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](True, z0, Cons(z1, z2)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), Cons(z1, z2)), MONUS(z0, Cons(z1, z2))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: @'_2, GT0_2, LGTH_1, EQLIST_2, GCD[ITE]_3, GCD_2, MONUS_2, GCD[FALSE][ITE]_3, MONUS[ITE]_3 Compound Symbols: c10_1, c13_1, c19_2, c21_1, c22_1, c6_2, c6_1, c18_2, c18_1, c26_3, c_1, c26, c8_2, c4_1, c1_1, c9_2 ---------------------------------------- (131) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace GCD[ITE](False, Cons(x0, x1), Cons(x2, x3)) -> c6(GT0(Cons(x0, x1), Cons(x2, x3))) by GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))), GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)), GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)), GT0(Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)), GT0(Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)), GT0(Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1)), GT0(Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](True, z0, Cons(z1, z2)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), Cons(z1, z2)), MONUS(z0, Cons(z1, z2))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: @'_2, GT0_2, LGTH_1, EQLIST_2, GCD[ITE]_3, GCD_2, MONUS_2, GCD[FALSE][ITE]_3, MONUS[ITE]_3 Compound Symbols: c10_1, c13_1, c19_2, c21_1, c22_1, c6_2, c18_2, c18_1, c26_3, c_1, c26, c8_2, c4_1, c1_1, c9_2, c6_1 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MONUS(x0, Cons(x1, x2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(x2))), Cons(Nil, Nil)), x0, Cons(x1, x2)), EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil)), LGTH(Cons(x1, x2))) by MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))), GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)), GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)), GT0(Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)), GT0(Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)), GT0(Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1)), GT0(Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](True, z0, Cons(z1, z2)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), Cons(z1, z2)), MONUS(z0, Cons(z1, z2))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) S tuples: @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: @'_2, GT0_2, LGTH_1, EQLIST_2, GCD[ITE]_3, GCD_2, MONUS_2, GCD[FALSE][ITE]_3, MONUS[ITE]_3 Compound Symbols: c10_1, c13_1, c19_2, c21_1, c22_1, c6_2, c18_2, c18_1, c26_3, c_1, c26, c8_2, c4_1, c1_1, c9_2, c6_1 ---------------------------------------- (135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace @'(Cons(z0, z1), z2) -> c10(@'(z1, z2)) by @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))), GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)), GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)), GT0(Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)), GT0(Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)), GT0(Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1)), GT0(Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](True, z0, Cons(z1, z2)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), Cons(z1, z2)), MONUS(z0, Cons(z1, z2))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) S tuples: GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GT0_2, LGTH_1, EQLIST_2, GCD[ITE]_3, GCD_2, MONUS_2, GCD[FALSE][ITE]_3, MONUS[ITE]_3, @'_2 Compound Symbols: c13_1, c19_2, c21_1, c22_1, c6_2, c18_2, c18_1, c26_3, c_1, c26, c8_2, c4_1, c1_1, c9_2, c6_1, c10_1 ---------------------------------------- (137) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))), GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)), GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)), GT0(Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)), GT0(Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)), GT0(Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1)), GT0(Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](True, z0, Cons(z1, z2)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), Cons(z1, z2)), MONUS(z0, Cons(z1, z2))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) S tuples: GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GT0_2, EQLIST_2, GCD[ITE]_3, GCD_2, MONUS_2, GCD[FALSE][ITE]_3, MONUS[ITE]_3, @'_2, LGTH_1 Compound Symbols: c13_1, c21_1, c22_1, c6_2, c18_2, c18_1, c26_3, c_1, c26, c8_2, c4_1, c1_1, c9_2, c6_1, c10_1, c19_1 ---------------------------------------- (139) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) We considered the (Usable) Rules:none And the Tuples: GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))), GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)), GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)), GT0(Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)), GT0(Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)), GT0(Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1)), GT0(Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](True, z0, Cons(z1, z2)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), Cons(z1, z2)), MONUS(z0, Cons(z1, z2))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(@(x_1, x_2)) = 0 POL(@'(x_1, x_2)) = x_1 POL(Cons(x_1, x_2)) = [1] + x_2 POL(EQLIST(x_1, x_2)) = 0 POL(False) = 0 POL(GCD(x_1, x_2)) = 0 POL(GCD[FALSE][ITE](x_1, x_2, x_3)) = 0 POL(GCD[ITE](x_1, x_2, x_3)) = 0 POL(GT0(x_1, x_2)) = 0 POL(LGTH(x_1)) = 0 POL(MONUS(x_1, x_2)) = 0 POL(MONUS[ITE](x_1, x_2, x_3)) = 0 POL(Nil) = 0 POL(True) = 0 POL(and(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c26) = 0 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2)) = x_1 + x_2 POL(eqList(x_1, x_2)) = 0 POL(gt0(x_1, x_2)) = [1] + x_1 + x_2 POL(lgth(x_1)) = 0 POL(monus(x_1, x_2)) = [1] + x_1 + x_2 POL(monus[Ite](x_1, x_2, x_3)) = [1] + x_3 ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))), GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)), GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)), GT0(Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)), GT0(Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)), GT0(Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1)), GT0(Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](True, z0, Cons(z1, z2)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), Cons(z1, z2)), MONUS(z0, Cons(z1, z2))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) S tuples: GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GT0_2, EQLIST_2, GCD[ITE]_3, GCD_2, MONUS_2, GCD[FALSE][ITE]_3, MONUS[ITE]_3, @'_2, LGTH_1 Compound Symbols: c13_1, c21_1, c22_1, c6_2, c18_2, c18_1, c26_3, c_1, c26, c8_2, c4_1, c1_1, c9_2, c6_1, c10_1, c19_1 ---------------------------------------- (141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3)) by GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))), GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)), GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)), GT0(Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)), GT0(Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)), GT0(Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1)), GT0(Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](True, z0, Cons(z1, z2)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), Cons(z1, z2)), MONUS(z0, Cons(z1, z2))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) S tuples: EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: EQLIST_2, GCD[ITE]_3, GCD_2, MONUS_2, GCD[FALSE][ITE]_3, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2 Compound Symbols: c21_1, c22_1, c6_2, c18_2, c18_1, c26_3, c_1, c26, c8_2, c4_1, c1_1, c9_2, c6_1, c10_1, c19_1, c13_1 ---------------------------------------- (143) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))), GT0(Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)), GT0(Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)), GT0(Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)), GT0(Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)), GT0(Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1)), GT0(Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](True, z0, Cons(z1, z2)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), Cons(z1, z2)), MONUS(z0, Cons(z1, z2))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) S tuples: EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: EQLIST_2, GCD[ITE]_3, GCD_2, MONUS_2, GCD[FALSE][ITE]_3, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2 Compound Symbols: c21_1, c22_1, c6_2, c18_2, c18_1, c26_3, c_1, c26, c8_2, c4_1, c1_1, c9_2, c6_1, c10_1, c19_1, c13_1 ---------------------------------------- (145) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing tuple parts ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[FALSE][ITE](True, z0, Cons(z1, z2)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), Cons(z1, z2)), MONUS(z0, Cons(z1, z2))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) S tuples: EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: EQLIST_2, GCD[ITE]_3, GCD_2, MONUS_2, GCD[FALSE][ITE]_3, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2 Compound Symbols: c21_1, c22_1, c6_2, c18_2, c18_1, c26_3, c_1, c26, c8_2, c4_1, c1_1, c9_2, c6_1, c10_1, c19_1, c13_1 ---------------------------------------- (147) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace GCD[FALSE][ITE](False, Cons(z0, z1), z2) -> c8(GCD(Cons(z0, z1), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z1))), Cons(Nil, Nil)), z2, Cons(z0, z1))), MONUS(z2, Cons(z0, z1))) by GCD[FALSE][ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c8(GCD(Cons(x0, Cons(x1, x2)), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x1, x2)))), Cons(Nil, Nil)), Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))), MONUS(Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[FALSE][ITE](True, z0, Cons(z1, z2)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), Cons(z1, z2)), MONUS(z0, Cons(z1, z2))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c8(GCD(Cons(x0, Cons(x1, x2)), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x1, x2)))), Cons(Nil, Nil)), Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))), MONUS(Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) S tuples: EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: EQLIST_2, GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, GCD[FALSE][ITE]_3, @'_2, LGTH_1, GT0_2 Compound Symbols: c21_1, c22_1, c6_2, c18_2, c18_1, c26_3, c_1, c26, c4_1, c1_1, c9_2, c6_1, c10_1, c19_1, c13_1, c8_2 ---------------------------------------- (149) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace GCD[FALSE][ITE](True, z0, Cons(z1, z2)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), Cons(z1, z2)), MONUS(z0, Cons(z1, z2))) by GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c8(GCD(Cons(x0, Cons(x1, x2)), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x1, x2)))), Cons(Nil, Nil)), Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))), MONUS(Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) S tuples: EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: EQLIST_2, GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3 Compound Symbols: c21_1, c22_1, c6_2, c18_2, c18_1, c26_3, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MONUS(x0, Cons(x1, Cons(z0, z1))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z1))), Cons(Nil, Nil)), x0, Cons(x1, Cons(z0, z1))), EQLIST(lgth(Cons(x1, Cons(z0, z1))), Cons(Nil, Nil)), LGTH(Cons(x1, Cons(z0, z1)))) by MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), EQLIST(@(Cons(Nil, Nil), lgth(Cons(z2, z3))), Cons(Nil, Nil)), LGTH(Cons(z1, Cons(z2, z3)))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c8(GCD(Cons(x0, Cons(x1, x2)), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x1, x2)))), Cons(Nil, Nil)), Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))), MONUS(Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), EQLIST(@(Cons(Nil, Nil), lgth(Cons(z2, z3))), Cons(Nil, Nil)), LGTH(Cons(z1, Cons(z2, z3)))) S tuples: EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), EQLIST(@(Cons(Nil, Nil), lgth(Cons(z2, z3))), Cons(Nil, Nil)), LGTH(Cons(z1, Cons(z2, z3)))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: EQLIST_2, GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3 Compound Symbols: c21_1, c22_1, c6_2, c18_2, c18_1, c_1, c26, c4_1, c1_1, c6_1, c26_3, c10_1, c19_1, c13_1, c8_2, c9_2 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(EQLIST(z0, z2)) by EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c8(GCD(Cons(x0, Cons(x1, x2)), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x1, x2)))), Cons(Nil, Nil)), Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))), MONUS(Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), EQLIST(@(Cons(Nil, Nil), lgth(Cons(z2, z3))), Cons(Nil, Nil)), LGTH(Cons(z1, Cons(z2, z3)))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) S tuples: EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), EQLIST(@(Cons(Nil, Nil), lgth(Cons(z2, z3))), Cons(Nil, Nil)), LGTH(Cons(z1, Cons(z2, z3)))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: EQLIST_2, GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3 Compound Symbols: c22_1, c6_2, c18_2, c18_1, c_1, c26, c4_1, c1_1, c6_1, c26_3, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(EQLIST(z1, z3)) by EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c8(GCD(Cons(x0, Cons(x1, x2)), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x1, x2)))), Cons(Nil, Nil)), Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))), MONUS(Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), EQLIST(@(Cons(Nil, Nil), lgth(Cons(z2, z3))), Cons(Nil, Nil)), LGTH(Cons(z1, Cons(z2, z3)))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c(EQLIST(lgth(Cons(x1, x2)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(EQLIST(lgth(Cons(x1, Nil)), Cons(Nil, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), EQLIST(@(Cons(Nil, Nil), lgth(z2)), Cons(Nil, Nil)), LGTH(Cons(z1, z2))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), EQLIST(@(Cons(Nil, Nil), lgth(Cons(z2, z3))), Cons(Nil, Nil)), LGTH(Cons(z1, Cons(z2, z3)))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c18_1, c_1, c26, c4_1, c1_1, c6_1, c26_3, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1 ---------------------------------------- (157) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c8(GCD(Cons(x0, Cons(x1, x2)), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x1, x2)))), Cons(Nil, Nil)), Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))), MONUS(Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c18_1, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2 ---------------------------------------- (159) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MONUS(x0, Cons(x1, Nil)) -> c1(MONUS[ITE](eqList(@(Cons(Nil, Nil), Nil), Cons(Nil, Nil)), x0, Cons(x1, Nil))) by MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c8(GCD(Cons(x0, Cons(x1, x2)), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x1, x2)))), Cons(Nil, Nil)), Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))), MONUS(Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c18_1, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GCD[FALSE][ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c8(GCD(Cons(x0, Cons(x1, x2)), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x1, x2)))), Cons(Nil, Nil)), Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))), MONUS(Cons(x3, Cons(x4, x5)), Cons(x0, Cons(x1, x2)))) by GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c18_1, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2 ---------------------------------------- (163) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace GCD(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)) -> c18(GCD[ITE](and(and(eqList(z0, z2), eqList(z1, z3)), eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Cons(z2, z3), x3))) by GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c18_1, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2 ---------------------------------------- (165) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace GCD(Cons(Nil, x1), Cons(Cons(z0, z1), x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Nil, x1), Cons(Cons(z0, z1), x3)), EQLIST(Cons(Nil, x1), Cons(Cons(z0, z1), x3))) by GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil)), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil)), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil)), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c18_1, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2 ---------------------------------------- (167) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c18_1, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2 ---------------------------------------- (169) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace GCD(Cons(Cons(z0, z1), x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(False, eqList(x1, x3)), Cons(Cons(z0, z1), x1), Cons(Nil, x3)), EQLIST(Cons(Cons(z0, z1), x1), Cons(Nil, x3))) by GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c18_1, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2 ---------------------------------------- (171) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c18_1, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2 ---------------------------------------- (173) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace GCD(Cons(Nil, x1), Cons(Nil, x3)) -> c18(GCD[ITE](and(True, eqList(x1, x3)), Cons(Nil, x1), Cons(Nil, x3)), EQLIST(Cons(Nil, x1), Cons(Nil, x3))) by GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1)), EQLIST(Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil)), EQLIST(Cons(Nil, z0), Cons(Nil, Nil))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1)), EQLIST(Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil)), EQLIST(Cons(Nil, z0), Cons(Nil, Nil))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1)), EQLIST(Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil)), EQLIST(Cons(Nil, z0), Cons(Nil, Nil))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c18_1, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2 ---------------------------------------- (175) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c18_1, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2 ---------------------------------------- (177) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Cons(x2, x3)), Cons(x0, Nil))), MONUS(Cons(x1, Cons(x2, x3)), Cons(x0, Nil))) by GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c18_1, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2 ---------------------------------------- (179) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace GCD(Cons(x0, x1), Cons(x2, x3)) -> c18(EQLIST(Cons(x0, x1), Cons(x2, x3))) by GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(Nil, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Nil, Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(Nil, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Nil, Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(Nil, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Nil, Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2, c18_1 ---------------------------------------- (181) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: GCD(Cons(Nil, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Nil, Nil), Cons(z2, z3))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2, c18_1 ---------------------------------------- (183) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c8(GCD(Cons(x0, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x1, Nil), Cons(x0, Nil))), MONUS(Cons(x1, Nil), Cons(x0, Nil))) by GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2, c18_1 ---------------------------------------- (185) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Cons(x2, x3), x4), Cons(Cons(x0, x1), Nil))) by GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2, c18_1 ---------------------------------------- (187) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), x5))) by GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), Cons(x2, x3))) -> c6(GT0(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), Cons(x2, x3)))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Nil)) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), Nil)) -> c6(GT0(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c6(GT0(Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Nil)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), Cons(x2, x3))) -> c6(GT0(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), Cons(x2, x3)))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Nil)) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), Nil)) -> c6(GT0(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c6(GT0(Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Nil)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2, c18_1 ---------------------------------------- (189) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing nodes: GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Nil)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), Nil)) -> c6(GT0(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Nil)) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4)) -> c6(GT0(Cons(Cons(x0, x1), Nil), Cons(Cons(x2, x3), x4))) GCD[ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), Cons(x2, x3))) -> c6(GT0(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), Cons(x2, x3)))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2, c18_1 ---------------------------------------- (191) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Cons(x0, x1), x2), Cons(Nil, Nil))), MONUS(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) by GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(z0, z1), z2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))), MONUS(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(z0, z1), z2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))), MONUS(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2, c18_1 ---------------------------------------- (193) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c8(GCD(Cons(Cons(x0, x1), Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x2), Cons(Cons(x0, x1), Nil))), MONUS(Cons(Nil, x2), Cons(Cons(x0, x1), Nil))) by GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, z2)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z2), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Nil, z2), Cons(Cons(z0, z1), Nil))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(z0, z1), z2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))), MONUS(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, z2)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z2), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Nil, z2), Cons(Cons(z0, z1), Nil))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c8_2, c9_2, c21_1, c22_1, c, c1, c26_2, c18_1 ---------------------------------------- (195) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(Nil, x0), Cons(Nil, Nil))), MONUS(Cons(Nil, x0), Cons(Nil, Nil))) by GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, z0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z0), Cons(Nil, Nil))), MONUS(Cons(Nil, z0), Cons(Nil, Nil))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(z0, z1), z2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))), MONUS(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, z2)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z2), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Nil, z2), Cons(Cons(z0, z1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, z0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z0), Cons(Nil, Nil))), MONUS(Cons(Nil, z0), Cons(Nil, Nil))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c9_2, c21_1, c22_1, c, c1, c26_2, c8_2, c18_1 ---------------------------------------- (197) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), x3)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), x3))) by GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(z1, z2), Cons(x2, x3))) -> c6(GT0(Cons(Nil, Nil), Cons(Cons(z1, z2), Cons(x2, x3)))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Nil)) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Nil))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(z1, z2), Nil)) -> c6(GT0(Cons(Nil, Nil), Cons(Cons(z1, z2), Nil))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4))) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c6(GT0(Cons(Nil, Nil), Cons(Cons(x0, x1), x2))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), Nil)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), Nil))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(z0, z1), z2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))), MONUS(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, z2)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z2), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Nil, z2), Cons(Cons(z0, z1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, z0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z0), Cons(Nil, Nil))), MONUS(Cons(Nil, z0), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(z1, z2), Cons(x2, x3))) -> c6(GT0(Cons(Nil, Nil), Cons(Cons(z1, z2), Cons(x2, x3)))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Nil)) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Nil))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(z1, z2), Nil)) -> c6(GT0(Cons(Nil, Nil), Cons(Cons(z1, z2), Nil))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4))) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c6(GT0(Cons(Nil, Nil), Cons(Cons(x0, x1), x2))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), Nil)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), Nil))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c9_2, c21_1, c22_1, c, c1, c26_2, c8_2, c18_1 ---------------------------------------- (199) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing nodes: GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(z1, z2), Cons(x2, x3))) -> c6(GT0(Cons(Nil, Nil), Cons(Cons(z1, z2), Cons(x2, x3)))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x0, x1), x2)) -> c6(GT0(Cons(Nil, Nil), Cons(Cons(x0, x1), x2))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), Nil)) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), Nil))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(z1, z2), Nil)) -> c6(GT0(Cons(Nil, Nil), Cons(Cons(z1, z2), Nil))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Nil)) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Nil))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(z0, z1), z2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))), MONUS(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, z2)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z2), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Nil, z2), Cons(Cons(z0, z1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, z0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z0), Cons(Nil, Nil))), MONUS(Cons(Nil, z0), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4))) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c9_2, c21_1, c22_1, c, c1, c26_2, c8_2, c18_1 ---------------------------------------- (201) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, x3)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, x3))) by GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, Cons(x2, x3))) -> c6(GT0(Cons(Cons(z0, z1), Nil), Cons(Nil, Cons(x2, x3)))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Nil)) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, Nil)) -> c6(GT0(Cons(Cons(z0, z1), Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c6(GT0(Cons(Cons(x0, x1), Nil), Cons(Nil, x2))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, Nil)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(z0, z1), z2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))), MONUS(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, z2)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z2), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Nil, z2), Cons(Cons(z0, z1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, z0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z0), Cons(Nil, Nil))), MONUS(Cons(Nil, z0), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4))) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, Cons(x2, x3))) -> c6(GT0(Cons(Cons(z0, z1), Nil), Cons(Nil, Cons(x2, x3)))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Nil)) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, Nil)) -> c6(GT0(Cons(Cons(z0, z1), Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c6(GT0(Cons(Cons(x0, x1), Nil), Cons(Nil, x2))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, Nil)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c9_2, c21_1, c22_1, c, c1, c26_2, c8_2, c18_1 ---------------------------------------- (203) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing nodes: GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, Nil)) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, Nil)) -> c6(GT0(Cons(Cons(z0, z1), Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, Cons(x2, x3))) -> c6(GT0(Cons(Cons(z0, z1), Nil), Cons(Nil, Cons(x2, x3)))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x2)) -> c6(GT0(Cons(Cons(x0, x1), Nil), Cons(Nil, x2))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Nil)) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Nil))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(z0, z1), z2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))), MONUS(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, z2)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z2), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Nil, z2), Cons(Cons(z0, z1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, z0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z0), Cons(Nil, Nil))), MONUS(Cons(Nil, z0), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4))) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c9_2, c21_1, c22_1, c, c1, c26_2, c8_2, c18_1 ---------------------------------------- (205) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Cons(x4, x5)))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))), Cons(x3, Cons(x4, x5))), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) by GCD[FALSE][ITE](True, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c9(GCD(monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z4, z5))), Nil)), Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))), Cons(z3, Cons(z4, z5))), MONUS(Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5)))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(z0, z1), z2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))), MONUS(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, z2)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z2), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Nil, z2), Cons(Cons(z0, z1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, z0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z0), Cons(Nil, Nil))), MONUS(Cons(Nil, z0), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4))) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4))) GCD[FALSE][ITE](True, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c9(GCD(monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z4, z5))), Nil)), Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))), Cons(z3, Cons(z4, z5))), MONUS(Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5)))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, GCD[FALSE][ITE]_3, EQLIST_2 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c9_2, c21_1, c22_1, c, c1, c26_2, c8_2, c18_1 ---------------------------------------- (207) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GCD[FALSE][ITE](True, Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c9(GCD(monus[Ite](eqList(Cons(Nil, @(Nil, lgth(Nil))), Cons(Nil, Nil)), Cons(x0, Cons(x1, x2)), Cons(x3, Nil)), Cons(x3, Nil)), MONUS(Cons(x0, Cons(x1, x2)), Cons(x3, Nil))) by GCD[FALSE][ITE](True, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c9(GCD(monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z0, Cons(z1, z2)), Cons(z3, Nil)), Cons(z3, Nil)), MONUS(Cons(z0, Cons(z1, z2)), Cons(z3, Nil))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(z0, z1), z2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))), MONUS(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, z2)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z2), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Nil, z2), Cons(Cons(z0, z1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, z0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z0), Cons(Nil, Nil))), MONUS(Cons(Nil, z0), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4))) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4))) GCD[FALSE][ITE](True, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c9(GCD(monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z4, z5))), Nil)), Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))), Cons(z3, Cons(z4, z5))), MONUS(Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5)))) GCD[FALSE][ITE](True, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c9(GCD(monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z0, Cons(z1, z2)), Cons(z3, Nil)), Cons(z3, Nil)), MONUS(Cons(z0, Cons(z1, z2)), Cons(z3, Nil))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, EQLIST_2, GCD[FALSE][ITE]_3 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c21_1, c22_1, c, c1, c26_2, c8_2, c18_1, c9_2 ---------------------------------------- (209) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace GCD[ITE](False, Cons(Nil, x0), Cons(Nil, x1)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, x1))) by GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Nil, Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Nil, Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, Cons(x2, x3))) -> c6(GT0(Cons(Nil, Nil), Cons(Nil, Cons(x2, x3)))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Nil, Nil)) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, Nil)) -> c6(GT0(Cons(Nil, Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, Cons(x1, x2))) -> c6(GT0(Cons(Nil, x0), Cons(Nil, Cons(x1, x2)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Nil, x2)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Nil, x2))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c6(GT0(Cons(Nil, Nil), Cons(Nil, x0))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, Nil)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, Nil))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(z0, z1), z2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))), MONUS(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, z2)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z2), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Nil, z2), Cons(Cons(z0, z1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, z0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z0), Cons(Nil, Nil))), MONUS(Cons(Nil, z0), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4))) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4))) GCD[FALSE][ITE](True, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c9(GCD(monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z4, z5))), Nil)), Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))), Cons(z3, Cons(z4, z5))), MONUS(Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5)))) GCD[FALSE][ITE](True, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c9(GCD(monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z0, Cons(z1, z2)), Cons(z3, Nil)), Cons(z3, Nil)), MONUS(Cons(z0, Cons(z1, z2)), Cons(z3, Nil))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Nil, Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Nil, Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, Cons(x2, x3))) -> c6(GT0(Cons(Nil, Nil), Cons(Nil, Cons(x2, x3)))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Nil, Nil)) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, Nil)) -> c6(GT0(Cons(Nil, Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, Cons(x1, x2))) -> c6(GT0(Cons(Nil, x0), Cons(Nil, Cons(x1, x2)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Nil, x2)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Nil, x2))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c6(GT0(Cons(Nil, Nil), Cons(Nil, x0))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, Nil)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, Nil))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, EQLIST_2, GCD[FALSE][ITE]_3 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c21_1, c22_1, c, c1, c26_2, c8_2, c18_1, c9_2 ---------------------------------------- (211) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing nodes: GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Nil, Nil)) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, Cons(x2, x3))) -> c6(GT0(Cons(Nil, Nil), Cons(Nil, Cons(x2, x3)))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, Nil)) -> c6(GT0(Cons(Nil, x0), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, Nil)) -> c6(GT0(Cons(Nil, Nil), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x0)) -> c6(GT0(Cons(Nil, Nil), Cons(Nil, x0))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(z0, z1), z2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))), MONUS(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, z2)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z2), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Nil, z2), Cons(Cons(z0, z1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, z0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z0), Cons(Nil, Nil))), MONUS(Cons(Nil, z0), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4))) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4))) GCD[FALSE][ITE](True, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c9(GCD(monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z4, z5))), Nil)), Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))), Cons(z3, Cons(z4, z5))), MONUS(Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5)))) GCD[FALSE][ITE](True, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c9(GCD(monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z0, Cons(z1, z2)), Cons(z3, Nil)), Cons(z3, Nil)), MONUS(Cons(z0, Cons(z1, z2)), Cons(z3, Nil))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Nil, Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Nil, Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, Cons(x1, x2))) -> c6(GT0(Cons(Nil, x0), Cons(Nil, Cons(x1, x2)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Nil, x2)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Nil, x2))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, EQLIST_2, GCD[FALSE][ITE]_3 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c21_1, c22_1, c, c1, c26_2, c8_2, c18_1, c9_2 ---------------------------------------- (213) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, lgth(z2))), Cons(Nil, Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) by MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](and(eqList(Nil, Nil), eqList(@(Nil, lgth(z2)), Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(z0, z1), z2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))), MONUS(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, z2)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z2), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Nil, z2), Cons(Cons(z0, z1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, z0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z0), Cons(Nil, Nil))), MONUS(Cons(Nil, z0), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4))) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4))) GCD[FALSE][ITE](True, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c9(GCD(monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z4, z5))), Nil)), Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))), Cons(z3, Cons(z4, z5))), MONUS(Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5)))) GCD[FALSE][ITE](True, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c9(GCD(monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z0, Cons(z1, z2)), Cons(z3, Nil)), Cons(z3, Nil)), MONUS(Cons(z0, Cons(z1, z2)), Cons(z3, Nil))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Nil, Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Nil, Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, Cons(x1, x2))) -> c6(GT0(Cons(Nil, x0), Cons(Nil, Cons(x1, x2)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Nil, x2)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Nil, x2))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](and(eqList(Nil, Nil), eqList(@(Nil, lgth(z2)), Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](and(eqList(Nil, Nil), eqList(@(Nil, lgth(z2)), Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, EQLIST_2, GCD[FALSE][ITE]_3 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c21_1, c22_1, c, c1, c26_2, c8_2, c18_1, c9_2 ---------------------------------------- (215) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(@(Cons(Nil, Nil), @(Cons(Nil, Nil), lgth(z3))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) by MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, @(Cons(Nil, Nil), lgth(z3)))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: gt0(Cons(z0, z1), Nil) -> True gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3) gt0(Nil, z0) -> False monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1) monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3) monus[Ite](True, Cons(z0, z1), z2) -> z1 eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3)) eqList(Nil, Cons(z0, z1)) -> False eqList(Cons(z0, z1), Nil) -> False eqList(Nil, Nil) -> True lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1)) lgth(Nil) -> Nil @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Tuples: GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c6(GCD[FALSE][ITE](gt0(z1, z3), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), GT0(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, y2))) -> c4(MONUS(z1, Cons(y1, y2))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Cons(y2, y3)))) -> c4(MONUS(z1, Cons(y1, Cons(y2, y3)))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Cons(y1, Nil))) -> c4(MONUS(z1, Cons(y1, Nil))) MONUS[ITE](False, Cons(z0, z1), Cons(z2, Nil)) -> c4(MONUS(z1, Nil)) MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) GCD[ITE](False, Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5))) -> c6(GT0(Cons(x0, Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) GCD[ITE](False, Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c6(GCD[FALSE][ITE](True, Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3))) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Cons(x2, x3)))) GCD[ITE](False, Cons(x0, Nil), Cons(x1, Nil)) -> c6(GCD[FALSE][ITE](False, Cons(x0, Nil), Cons(x1, Nil))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Cons(x3, x4), x5))) GCD[ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(x1, x2), x3))) GCD[ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3)) -> c6(GCD[FALSE][ITE](False, Cons(Cons(x0, x1), Nil), Cons(Nil, x3))) GCD[ITE](False, Cons(Nil, Nil), Cons(Nil, x1)) -> c6(GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, x1))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD[FALSE][ITE](False, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c8(GCD(Cons(z0, Cons(z1, z2)), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z1, z2))), Nil)), Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))), MONUS(Cons(z3, Cons(z4, z5)), Cons(z0, Cons(z1, z2)))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Cons(z2, z3)), Cons(z0, Nil))), MONUS(Cons(z1, Cons(z2, z3)), Cons(z0, Nil))) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) GCD[FALSE][ITE](False, Cons(z0, Nil), Cons(z1, Nil)) -> c8(GCD(Cons(z0, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z1, Nil), Cons(z0, Nil))), MONUS(Cons(z1, Nil), Cons(z0, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Cons(z2, z3), z4)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Cons(z2, z3), z4), Cons(Cons(z0, z1), Nil))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Cons(x3, x4), Cons(x5, x6)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Cons(x4, x5), x6))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Cons(z0, z1), z2)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))), MONUS(Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD[FALSE][ITE](False, Cons(Cons(z0, z1), Nil), Cons(Nil, z2)) -> c8(GCD(Cons(Cons(z0, z1), Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z2), Cons(Cons(z0, z1), Nil))), MONUS(Cons(Nil, z2), Cons(Cons(z0, z1), Nil))) GCD[FALSE][ITE](False, Cons(Nil, Nil), Cons(Nil, z0)) -> c8(GCD(Cons(Nil, Nil), monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(Nil, z0), Cons(Nil, Nil))), MONUS(Cons(Nil, z0), Cons(Nil, Nil))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4))) -> c6(GT0(Cons(Nil, x0), Cons(Cons(x1, x2), Cons(x3, x4)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Cons(x2, x3), x4))) GCD[ITE](False, Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5))) -> c6(GT0(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, Cons(x4, x5)))) GCD[ITE](False, Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4))) -> c6(GT0(Cons(Cons(x0, x1), x2), Cons(Nil, Cons(x3, x4)))) GCD[ITE](False, Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4)) -> c6(GT0(Cons(Cons(x0, x1), Cons(x2, x3)), Cons(Nil, x4))) GCD[FALSE][ITE](True, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c9(GCD(monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Cons(z4, z5))), Nil)), Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))), Cons(z3, Cons(z4, z5))), MONUS(Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5)))) GCD[FALSE][ITE](True, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c9(GCD(monus[Ite](and(eqList(Nil, Nil), eqList(@(Nil, lgth(Nil)), Nil)), Cons(z0, Cons(z1, z2)), Cons(z3, Nil)), Cons(z3, Nil)), MONUS(Cons(z0, Cons(z1, z2)), Cons(z3, Nil))) GCD[ITE](False, Cons(Nil, Cons(x1, x2)), Cons(Nil, Cons(x4, x5))) -> c6(GT0(Cons(Nil, Cons(x1, x2)), Cons(Nil, Cons(x4, x5)))) GCD[ITE](False, Cons(Nil, x0), Cons(Nil, Cons(x1, x2))) -> c6(GT0(Cons(Nil, x0), Cons(Nil, Cons(x1, x2)))) GCD[ITE](False, Cons(Nil, Cons(x0, x1)), Cons(Nil, x2)) -> c6(GT0(Cons(Nil, Cons(x0, x1)), Cons(Nil, x2))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](and(eqList(Nil, Nil), eqList(@(Nil, lgth(z2)), Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, @(Cons(Nil, Nil), lgth(z3)))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) S tuples: GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c18(GCD[ITE](and(eqList(x0, x2), and(eqList(z0, z2), eqList(z1, z3))), Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3)))) GCD(Cons(x0, Nil), Cons(x2, Cons(z0, z1))) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Nil), Cons(x2, Cons(z0, z1))), EQLIST(Cons(x0, Nil), Cons(x2, Cons(z0, z1)))) GCD(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), False), Cons(x0, Cons(z0, z1)), Cons(x2, Nil)), EQLIST(Cons(x0, Cons(z0, z1)), Cons(x2, Nil))) GCD(Cons(x0, Nil), Cons(x2, Nil)) -> c18(GCD[ITE](and(eqList(x0, x2), True), Cons(x0, Nil), Cons(x2, Nil)), EQLIST(Cons(x0, Nil), Cons(x2, Nil))) MONUS(x0, Cons(x1, x2)) -> c(LGTH(Cons(x1, x2))) MONUS(x0, Nil) -> c26 MONUS(x0, Cons(x1, Nil)) -> c1(LGTH(Cons(x1, Nil))) LGTH(Cons(z0, z1)) -> c19(LGTH(z1)) GT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c13(GT0(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(Cons(y0, y1), z1), Cons(Cons(y2, y3), z3)) -> c21(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c22(EQLIST(Cons(y0, y1), Cons(y2, y3))) EQLIST(Cons(z0, Cons(Cons(y0, y1), y2)), Cons(z2, Cons(Cons(y3, y4), y5))) -> c22(EQLIST(Cons(Cons(y0, y1), y2), Cons(Cons(y3, y4), y5))) MONUS(x0, Cons(x1, x2)) -> c MONUS(x0, Cons(x1, Nil)) -> c1 MONUS(z0, Cons(z1, Nil)) -> c1(MONUS[ITE](eqList(Cons(Nil, @(Nil, Nil)), Cons(Nil, Nil)), z0, Cons(z1, Nil))) GCD(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Nil, z5)), Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Nil), Cons(Cons(z3, z4), z5))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil)), EQLIST(Cons(Cons(z0, z1), z2), Cons(Cons(z3, z4), Nil))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)) -> c18(GCD[ITE](and(and(eqList(z0, z3), eqList(z1, z4)), eqList(Cons(x1, x2), z5)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Cons(z3, z4), z5))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Cons(z1, z2), Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, Nil), Cons(Cons(z1, z2), z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Nil, Nil), Cons(Cons(z1, z2), z3))) GCD(Cons(Nil, z0), Cons(Cons(z1, z2), Nil)) -> c18(GCD[ITE](and(False, eqList(z0, Nil)), Cons(Nil, z0), Cons(Cons(z1, z2), Nil))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(False, eqList(z2, Cons(x4, x5))), Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Cons(z0, z1), z2), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Cons(x1, x2), z3)), Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3)), EQLIST(Cons(Cons(z0, z1), Cons(x1, x2)), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), Nil), Cons(Nil, z3)) -> c18(GCD[ITE](and(False, eqList(Nil, z3)), Cons(Cons(z0, z1), Nil), Cons(Nil, z3))) GCD(Cons(Cons(z0, z1), z2), Cons(Nil, Nil)) -> c18(GCD[ITE](and(False, eqList(z2, Nil)), Cons(Cons(z0, z1), z2), Cons(Nil, Nil))) GCD(Cons(Nil, z0), Cons(Nil, Cons(x4, x5))) -> c18(GCD[ITE](and(True, eqList(z0, Cons(x4, x5))), Cons(Nil, z0), Cons(Nil, Cons(x4, x5))), EQLIST(Cons(Nil, z0), Cons(Nil, Cons(x4, x5)))) GCD(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Cons(x1, x2), z1)), Cons(Nil, Cons(x1, x2)), Cons(Nil, z1)), EQLIST(Cons(Nil, Cons(x1, x2)), Cons(Nil, z1))) GCD(Cons(Nil, Nil), Cons(Nil, z1)) -> c18(GCD[ITE](and(True, eqList(Nil, z1)), Cons(Nil, Nil), Cons(Nil, z1))) GCD(Cons(Nil, z0), Cons(Nil, Nil)) -> c18(GCD[ITE](and(True, eqList(z0, Nil)), Cons(Nil, z0), Cons(Nil, Nil))) MONUS(z0, Cons(z1, z2)) -> c26(MONUS[ITE](and(eqList(Nil, Nil), eqList(@(Nil, lgth(z2)), Nil)), z0, Cons(z1, z2)), LGTH(Cons(z1, z2))) MONUS(z0, Cons(z1, Cons(z2, z3))) -> c26(MONUS[ITE](eqList(Cons(Nil, @(Nil, @(Cons(Nil, Nil), lgth(z3)))), Cons(Nil, Nil)), z0, Cons(z1, Cons(z2, z3))), LGTH(Cons(z1, Cons(z2, z3)))) K tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c10(@'(Cons(y0, y1), z2)) GCD(Cons(x0, Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Nil), Cons(z2, z3))) GCD(Cons(Cons(x0, x1), Nil), Cons(z2, z3)) -> c18(EQLIST(Cons(Cons(x0, x1), Nil), Cons(z2, z3))) GCD(Cons(z0, z1), Cons(x3, Cons(x4, x5))) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Cons(x4, x5)))) GCD(Cons(z0, z1), Cons(x3, Nil)) -> c18(EQLIST(Cons(z0, z1), Cons(x3, Nil))) GCD(Cons(x0, Cons(x1, x2)), Cons(z2, z3)) -> c18(EQLIST(Cons(x0, Cons(x1, x2)), Cons(z2, z3))) Defined Rule Symbols: gt0_2, monus_2, monus[Ite]_3, eqList_2, lgth_1, @_2, and_2 Defined Pair Symbols: GCD[ITE]_3, GCD_2, MONUS_2, MONUS[ITE]_3, @'_2, LGTH_1, GT0_2, EQLIST_2, GCD[FALSE][ITE]_3 Compound Symbols: c6_2, c18_2, c_1, c26, c4_1, c1_1, c6_1, c10_1, c19_1, c13_1, c21_1, c22_1, c, c1, c8_2, c18_1, c9_2, c26_2