KILLED proof of input_9ETOQ1HU4y.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).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), 188 ms] (2) CpxRelTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (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), 0 ms] (16) CpxRNTS (17) InliningProof [UPPER BOUND(ID), 560 ms] (18) CpxRNTS (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 1 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 50 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 6 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 298 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 47 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 379 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 47 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 1185 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 479 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 3044 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 973 ms] (52) CpxRNTS (53) CompletionProof [UPPER BOUND(ID), 0 ms] (54) CpxTypedWeightedCompleteTrs (55) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (58) CdtProblem (59) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 137 ms] (66) CdtProblem (67) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 40 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 25 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 16 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms] (192) CdtProblem (193) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 1 ms] (212) CdtProblem (213) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (218) CdtProblem (219) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (220) CdtProblem (221) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CdtProblem (223) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CdtProblem (225) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 40 ms] (226) CdtProblem (227) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 57 ms] (228) CdtProblem (229) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 99 ms] (230) CdtProblem (231) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 279 ms] (232) 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: lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) lt0(x, Nil) -> False g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) f(x, Cons(x', xs)) -> f(f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)), f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))) number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil)) The (relative) TRS S consists of the following rules: g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y)) g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs) f[Ite][False][Ite](False, Cons(x, xs), y) -> xs f[Ite][False][Ite](True, x', Cons(x, xs)) -> xs f[Ite][False][Ite](False, x, y) -> Cons(Cons(Nil, Nil), y) f[Ite][False][Ite](True, x, y) -> x 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: lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) lt0(x, Nil) -> False g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) f(x, Cons(x', xs)) -> f(f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)), f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))) number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil)) The (relative) TRS S consists of the following rules: g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y)) g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs) f[Ite][False][Ite](False, Cons(x, xs), y) -> xs f[Ite][False][Ite](True, x', Cons(x, xs)) -> xs f[Ite][False][Ite](False, x, y) -> Cons(Cons(Nil, Nil), y) f[Ite][False][Ite](True, x, y) -> x Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) lt0(x, Nil) -> False g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) f(x, Cons(x', xs)) -> f(f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)), f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))) number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil)) g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y)) g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs) f[Ite][False][Ite](False, Cons(x, xs), y) -> xs f[Ite][False][Ite](True, x', Cons(x, xs)) -> xs f[Ite][False][Ite](False, x, y) -> Cons(Cons(Nil, Nil), y) f[Ite][False][Ite](True, x, y) -> x S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) 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: lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) lt0(x, Nil) -> False g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) f(x, Cons(x', xs)) -> f(f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)), f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))) number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil)) The (relative) TRS S consists of the following rules: g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y)) g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs) f[Ite][False][Ite](False, Cons(x, xs), y) -> xs f[Ite][False][Ite](True, x', Cons(x, xs)) -> xs f[Ite][False][Ite](False, x, y) -> Cons(Cons(Nil, Nil), y) f[Ite][False][Ite](True, x, y) -> x 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: lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) [1] g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] lt0(x, Nil) -> False [1] g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1] f(x, Cons(x', xs)) -> f(f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)), f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))) [1] number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil)) [1] g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y)) [0] g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs) [0] f[Ite][False][Ite](False, Cons(x, xs), y) -> xs [0] f[Ite][False][Ite](True, x', Cons(x, xs)) -> xs [0] f[Ite][False][Ite](False, x, y) -> Cons(Cons(Nil, Nil), y) [0] f[Ite][False][Ite](True, x, y) -> x [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: lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) [1] g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] lt0(x, Nil) -> False [1] g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1] f(x, Cons(x', xs)) -> f(f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)), f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))) [1] number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil)) [1] g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y)) [0] g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs) [0] f[Ite][False][Ite](False, Cons(x, xs), y) -> xs [0] f[Ite][False][Ite](True, x', Cons(x, xs)) -> xs [0] f[Ite][False][Ite](False, x, y) -> Cons(Cons(Nil, Nil), y) [0] f[Ite][False][Ite](True, x, y) -> x [0] The TRS has the following type information: lt0 :: Cons:Nil -> Cons:Nil -> False:True Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil g :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil f :: Cons:Nil -> Cons:Nil -> Cons:Nil False :: False:True g[Ite][False][Ite] :: False:True -> Cons:Nil -> Cons:Nil -> Cons:Nil f[Ite][False][Ite] :: False:True -> Cons:Nil -> Cons:Nil -> Cons:Nil number42 :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil True :: False:True 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: g_2 f_2 number42 goal_2 (c) The following functions are completely defined: lt0_2 g[Ite][False][Ite]_3 f[Ite][False][Ite]_3 Due to the following rules being added: g[Ite][False][Ite](v0, v1, v2) -> Nil [0] f[Ite][False][Ite](v0, v1, v2) -> Nil [0] lt0(v0, v1) -> null_lt0 [0] And the following fresh constants: null_lt0 ---------------------------------------- (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: lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) [1] g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] lt0(x, Nil) -> False [1] g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1] f(x, Cons(x', xs)) -> f(f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)), f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))) [1] number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil)) [1] g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y)) [0] g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs) [0] f[Ite][False][Ite](False, Cons(x, xs), y) -> xs [0] f[Ite][False][Ite](True, x', Cons(x, xs)) -> xs [0] f[Ite][False][Ite](False, x, y) -> Cons(Cons(Nil, Nil), y) [0] f[Ite][False][Ite](True, x, y) -> x [0] g[Ite][False][Ite](v0, v1, v2) -> Nil [0] f[Ite][False][Ite](v0, v1, v2) -> Nil [0] lt0(v0, v1) -> null_lt0 [0] The TRS has the following type information: lt0 :: Cons:Nil -> Cons:Nil -> False:True:null_lt0 Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil g :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil f :: Cons:Nil -> Cons:Nil -> Cons:Nil False :: False:True:null_lt0 g[Ite][False][Ite] :: False:True:null_lt0 -> Cons:Nil -> Cons:Nil -> Cons:Nil f[Ite][False][Ite] :: False:True:null_lt0 -> Cons:Nil -> Cons:Nil -> Cons:Nil number42 :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil True :: False:True:null_lt0 null_lt0 :: False:True:null_lt0 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: lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) [1] g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] lt0(x, Nil) -> False [1] g(Cons(x'', xs''), Cons(x', xs)) -> g[Ite][False][Ite](lt0(xs'', Nil), Cons(x'', xs''), Cons(x', xs)) [2] g(x, Cons(x', xs)) -> g[Ite][False][Ite](null_lt0, x, Cons(x', xs)) [1] f(Cons(x''', xs'''), Cons(x', xs)) -> f(f[Ite][False][Ite](lt0(xs''', Nil), Cons(x''', xs'''), Cons(x', xs)), f[Ite][False][Ite](lt0(xs''', Nil), Cons(x''', xs'''), Cons(x', xs))) [3] f(Cons(x''', xs'''), Cons(x', xs)) -> f(f[Ite][False][Ite](lt0(xs''', Nil), Cons(x''', xs'''), Cons(x', xs)), f[Ite][False][Ite](null_lt0, Cons(x''', xs'''), Cons(x', xs))) [2] f(Cons(x'1, xs'1), Cons(x', xs)) -> f(f[Ite][False][Ite](null_lt0, Cons(x'1, xs'1), Cons(x', xs)), f[Ite][False][Ite](lt0(xs'1, Nil), Cons(x'1, xs'1), Cons(x', xs))) [2] f(x, Cons(x', xs)) -> f(f[Ite][False][Ite](null_lt0, x, Cons(x', xs)), f[Ite][False][Ite](null_lt0, x, Cons(x', xs))) [1] number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil)) [1] g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y)) [0] g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs) [0] f[Ite][False][Ite](False, Cons(x, xs), y) -> xs [0] f[Ite][False][Ite](True, x', Cons(x, xs)) -> xs [0] f[Ite][False][Ite](False, x, y) -> Cons(Cons(Nil, Nil), y) [0] f[Ite][False][Ite](True, x, y) -> x [0] g[Ite][False][Ite](v0, v1, v2) -> Nil [0] f[Ite][False][Ite](v0, v1, v2) -> Nil [0] lt0(v0, v1) -> null_lt0 [0] The TRS has the following type information: lt0 :: Cons:Nil -> Cons:Nil -> False:True:null_lt0 Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil g :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil f :: Cons:Nil -> Cons:Nil -> Cons:Nil False :: False:True:null_lt0 g[Ite][False][Ite] :: False:True:null_lt0 -> Cons:Nil -> Cons:Nil -> Cons:Nil f[Ite][False][Ite] :: False:True:null_lt0 -> Cons:Nil -> Cons:Nil -> Cons:Nil number42 :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil True :: False:True:null_lt0 null_lt0 :: False:True:null_lt0 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 False => 1 True => 2 null_lt0 => 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), f[Ite][False][Ite](0, 1 + x''' + xs''', 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 1 }-> f(f[Ite][False][Ite](0, x, 1 + x' + xs), f[Ite][False][Ite](0, x, 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x >= 0, x' >= 0, z = x f(z, z') -{ 2 }-> f(f[Ite][False][Ite](0, 1 + x'1 + xs'1, 1 + x' + xs), f[Ite][False][Ite](lt0(xs'1, 0), 1 + x'1 + xs'1, 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: x >= 0, z = x, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> x :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z'' = y, z = 1, x >= 0, y >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + y :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, x, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x >= 0, x' >= 0, z = x g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: x >= 0, z = x, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(x', xs) :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + y) :|: xs >= 0, z' = 1 + x + xs, z'' = y, z = 1, x >= 0, y >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 goal(z, z') -{ 1 }-> 1 + f(x, y) + (1 + g(x, y) + 0) :|: x >= 0, y >= 0, z = x, z' = y lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: ---------------------------------------- (17) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> x :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z'' = y, z = 1, x >= 0, y >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + y :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), 0) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 2 }-> f(0, f[Ite][False][Ite](lt0(xs'1, 0), 1 + x'1 + xs'1, 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, x >= 0, x' >= 0, z = x, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, x = v1, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', v1' >= 0, 0 = v0', x = v1', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: x >= 0, z = x, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> x :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z'' = y, z = 1, x >= 0, y >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + y :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, x, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x >= 0, x' >= 0, z = x g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: x >= 0, z = x, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(x', xs) :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + y) :|: xs >= 0, z' = 1 + x + xs, z'' = y, z = 1, x >= 0, y >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 goal(z, z') -{ 1 }-> 1 + f(x, y) + (1 + g(x, y) + 0) :|: x >= 0, y >= 0, z = x, z' = y lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 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: f(z, z') -{ 3 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), 0) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 2 }-> f(0, f[Ite][False][Ite](lt0(xs'1, 0), 1 + x'1 + xs'1, 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, z, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: ---------------------------------------- (21) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { number42 } { lt0 } { f[Ite][False][Ite] } { g[Ite][False][Ite], g } { f } { goal } ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), 0) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 2 }-> f(0, f[Ite][False][Ite](lt0(xs'1, 0), 1 + x'1 + xs'1, 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, z, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {number42}, {lt0}, {f[Ite][False][Ite]}, {g[Ite][False][Ite],g}, {f}, {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: f(z, z') -{ 3 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), 0) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 2 }-> f(0, f[Ite][False][Ite](lt0(xs'1, 0), 1 + x'1 + xs'1, 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, z, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {number42}, {lt0}, {f[Ite][False][Ite]}, {g[Ite][False][Ite],g}, {f}, {goal} ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: number42 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 42 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), 0) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 2 }-> f(0, f[Ite][False][Ite](lt0(xs'1, 0), 1 + x'1 + xs'1, 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, z, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {number42}, {lt0}, {f[Ite][False][Ite]}, {g[Ite][False][Ite],g}, {f}, {goal} Previous analysis results are: number42: runtime: ?, size: O(1) [42] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: number42 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), 0) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 2 }-> f(0, f[Ite][False][Ite](lt0(xs'1, 0), 1 + x'1 + xs'1, 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, z, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {lt0}, {f[Ite][False][Ite]}, {g[Ite][False][Ite],g}, {f}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] ---------------------------------------- (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: f(z, z') -{ 3 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), 0) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 2 }-> f(0, f[Ite][False][Ite](lt0(xs'1, 0), 1 + x'1 + xs'1, 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, z, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {lt0}, {f[Ite][False][Ite]}, {g[Ite][False][Ite],g}, {f}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lt0 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), 0) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 2 }-> f(0, f[Ite][False][Ite](lt0(xs'1, 0), 1 + x'1 + xs'1, 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, z, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {lt0}, {f[Ite][False][Ite]}, {g[Ite][False][Ite],g}, {f}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] lt0: runtime: ?, size: O(1) [1] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: lt0 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 3 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 2 }-> f(f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs), 0) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 2 }-> f(0, f[Ite][False][Ite](lt0(xs'1, 0), 1 + x'1 + xs'1, 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 2 }-> g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, z, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {f[Ite][False][Ite]}, {g[Ite][False][Ite],g}, {f}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] lt0: runtime: O(n^1) [1 + z'], size: O(1) [1] ---------------------------------------- (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: f(z, z') -{ 5 }-> f(f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs), f[Ite][False][Ite](s1, 1 + x''' + xs''', 1 + x' + xs)) :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 3 }-> f(f[Ite][False][Ite](s2, 1 + x''' + xs''', 1 + x' + xs), 0) :|: s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 3 }-> f(0, f[Ite][False][Ite](s3, 1 + x'1 + xs'1, 1 + x' + xs)) :|: s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 3 }-> g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, z, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {f[Ite][False][Ite]}, {g[Ite][False][Ite],g}, {f}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] lt0: runtime: O(n^1) [1 + z'], size: O(1) [1] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f[Ite][False][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' + z'' ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 }-> f(f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs), f[Ite][False][Ite](s1, 1 + x''' + xs''', 1 + x' + xs)) :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 3 }-> f(f[Ite][False][Ite](s2, 1 + x''' + xs''', 1 + x' + xs), 0) :|: s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 3 }-> f(0, f[Ite][False][Ite](s3, 1 + x'1 + xs'1, 1 + x' + xs)) :|: s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 3 }-> g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, z, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {f[Ite][False][Ite]}, {g[Ite][False][Ite],g}, {f}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] lt0: runtime: O(n^1) [1 + z'], size: O(1) [1] f[Ite][False][Ite]: runtime: ?, size: O(n^1) [2 + z' + z''] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f[Ite][False][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 }-> f(f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs), f[Ite][False][Ite](s1, 1 + x''' + xs''', 1 + x' + xs)) :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 3 }-> f(f[Ite][False][Ite](s2, 1 + x''' + xs''', 1 + x' + xs), 0) :|: s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 3 }-> f(0, f[Ite][False][Ite](s3, 1 + x'1 + xs'1, 1 + x' + xs)) :|: s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 3 }-> g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, z, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {g[Ite][False][Ite],g}, {f}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] lt0: runtime: O(n^1) [1 + z'], size: O(1) [1] f[Ite][False][Ite]: runtime: O(1) [0], size: O(n^1) [2 + 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: f(z, z') -{ 5 }-> f(s4, s5) :|: s4 >= 0, s4 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s5 >= 0, s5 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 3 }-> f(s6, 0) :|: s6 >= 0, s6 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 3 }-> f(0, s7) :|: s7 >= 0, s7 <= 1 + x' + xs + 2 + (1 + x'1 + xs'1), s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 3 }-> g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, z, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {g[Ite][False][Ite],g}, {f}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] lt0: runtime: O(n^1) [1 + z'], size: O(1) [1] f[Ite][False][Ite]: runtime: O(1) [0], size: O(n^1) [2 + z' + z''] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: g[Ite][False][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 42 Computed SIZE bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 42 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 }-> f(s4, s5) :|: s4 >= 0, s4 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s5 >= 0, s5 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 3 }-> f(s6, 0) :|: s6 >= 0, s6 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 3 }-> f(0, s7) :|: s7 >= 0, s7 <= 1 + x' + xs + 2 + (1 + x'1 + xs'1), s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 3 }-> g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, z, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {g[Ite][False][Ite],g}, {f}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] lt0: runtime: O(n^1) [1 + z'], size: O(1) [1] f[Ite][False][Ite]: runtime: O(1) [0], size: O(n^1) [2 + z' + z''] g[Ite][False][Ite]: runtime: ?, size: O(1) [42] g: runtime: ?, size: O(1) [42] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: g[Ite][False][Ite] after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 7 + 3*z' Computed RUNTIME bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 10 + 3*z ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 }-> f(s4, s5) :|: s4 >= 0, s4 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s5 >= 0, s5 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 3 }-> f(s6, 0) :|: s6 >= 0, s6 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 3 }-> f(0, s7) :|: s7 >= 0, s7 <= 1 + x' + xs + 2 + (1 + x'1 + xs'1), s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 3 }-> g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](0, z, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 1 }-> 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0 lt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {f}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] lt0: runtime: O(n^1) [1 + z'], size: O(1) [1] f[Ite][False][Ite]: runtime: O(1) [0], size: O(n^1) [2 + z' + z''] g[Ite][False][Ite]: runtime: O(n^1) [7 + 3*z'], size: O(1) [42] g: runtime: O(n^1) [10 + 3*z], size: O(1) [42] ---------------------------------------- (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: f(z, z') -{ 5 }-> f(s4, s5) :|: s4 >= 0, s4 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s5 >= 0, s5 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 3 }-> f(s6, 0) :|: s6 >= 0, s6 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 3 }-> f(0, s7) :|: s7 >= 0, s7 <= 1 + x' + xs + 2 + (1 + x'1 + xs'1), s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 13 + 3*x'' + 3*xs'' }-> s8 :|: s8 >= 0, s8 <= 42, s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 8 + 3*z }-> s9 :|: s9 >= 0, s9 <= 42, xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 10 + 3*xs }-> s11 :|: s11 >= 0, s11 <= 42, xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 10 + 3*z' }-> s12 :|: s12 >= 0, s12 <= 42, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 11 + 3*z }-> 1 + f(z, z') + (1 + s10 + 0) :|: s10 >= 0, s10 <= 42, z >= 0, z' >= 0 lt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {f}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] lt0: runtime: O(n^1) [1 + z'], size: O(1) [1] f[Ite][False][Ite]: runtime: O(1) [0], size: O(n^1) [2 + z' + z''] g[Ite][False][Ite]: runtime: O(n^1) [7 + 3*z'], size: O(1) [42] g: runtime: O(n^1) [10 + 3*z], size: O(1) [42] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 42 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 }-> f(s4, s5) :|: s4 >= 0, s4 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s5 >= 0, s5 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 3 }-> f(s6, 0) :|: s6 >= 0, s6 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 3 }-> f(0, s7) :|: s7 >= 0, s7 <= 1 + x' + xs + 2 + (1 + x'1 + xs'1), s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 13 + 3*x'' + 3*xs'' }-> s8 :|: s8 >= 0, s8 <= 42, s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 8 + 3*z }-> s9 :|: s9 >= 0, s9 <= 42, xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 10 + 3*xs }-> s11 :|: s11 >= 0, s11 <= 42, xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 10 + 3*z' }-> s12 :|: s12 >= 0, s12 <= 42, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 11 + 3*z }-> 1 + f(z, z') + (1 + s10 + 0) :|: s10 >= 0, s10 <= 42, z >= 0, z' >= 0 lt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {f}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] lt0: runtime: O(n^1) [1 + z'], size: O(1) [1] f[Ite][False][Ite]: runtime: O(1) [0], size: O(n^1) [2 + z' + z''] g[Ite][False][Ite]: runtime: O(n^1) [7 + 3*z'], size: O(1) [42] g: runtime: O(n^1) [10 + 3*z], size: O(1) [42] f: runtime: ?, size: O(1) [42] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 }-> f(s4, s5) :|: s4 >= 0, s4 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s5 >= 0, s5 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''' f(z, z') -{ 3 }-> f(s6, 0) :|: s6 >= 0, s6 <= 1 + x' + xs + 2 + (1 + x''' + xs'''), s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs''', v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x''' + xs''' = v1, v2 >= 0 f(z, z') -{ 3 }-> f(0, s7) :|: s7 >= 0, s7 <= 1 + x' + xs + 2 + (1 + x'1 + xs'1), s3 >= 0, s3 <= 1, xs >= 0, z' = 1 + x' + xs, x'1 >= 0, x' >= 0, z = 1 + x'1 + xs'1, xs'1 >= 0, v0 >= 0, 1 + x' + xs = v2, v1 >= 0, 0 = v0, 1 + x'1 + xs'1 = v1, v2 >= 0 f(z, z') -{ 1 }-> f(0, 0) :|: xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0, v0 >= 0, 1 + x' + xs = v2, 0 = v0, v2 >= 0, v0' >= 0, 1 + x' + xs = v2', 0 = v0', v2' >= 0 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> z' :|: z = 2, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + z'' :|: z = 1, z' >= 0, z'' >= 0 g(z, z') -{ 13 + 3*x'' + 3*xs'' }-> s8 :|: s8 >= 0, s8 <= 42, s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0 g(z, z') -{ 8 + 3*z }-> s9 :|: s9 >= 0, s9 <= 42, xs >= 0, z' = 1 + x' + xs, z >= 0, x' >= 0 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 10 + 3*xs }-> s11 :|: s11 >= 0, s11 <= 42, xs >= 0, z' = 1 + x + xs, z = 1, x >= 0, z'' >= 0 g[Ite][False][Ite](z, z', z'') -{ 10 + 3*z' }-> s12 :|: s12 >= 0, s12 <= 42, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 goal(z, z') -{ 11 + 3*z }-> 1 + f(z, z') + (1 + s10 + 0) :|: s10 >= 0, s10 <= 42, z >= 0, z' >= 0 lt0(z, z') -{ 2 + xs }-> s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Function symbols to be analyzed: {f}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] lt0: runtime: O(n^1) [1 + z'], size: O(1) [1] f[Ite][False][Ite]: runtime: O(1) [0], size: O(n^1) [2 + z' + z''] g[Ite][False][Ite]: runtime: O(n^1) [7 + 3*z'], size: O(1) [42] g: runtime: O(n^1) [10 + 3*z], size: O(1) [42] f: runtime: INF, size: O(1) [42] ---------------------------------------- (53) 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: g[Ite][False][Ite](v0, v1, v2) -> null_g[Ite][False][Ite] [0] f[Ite][False][Ite](v0, v1, v2) -> null_f[Ite][False][Ite] [0] lt0(v0, v1) -> null_lt0 [0] g(v0, v1) -> null_g [0] f(v0, v1) -> null_f [0] And the following fresh constants: null_g[Ite][False][Ite], null_f[Ite][False][Ite], null_lt0, null_g, null_f ---------------------------------------- (54) 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: lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) [1] g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] lt0(x, Nil) -> False [1] g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1] f(x, Cons(x', xs)) -> f(f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)), f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))) [1] number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil)) [1] g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y)) [0] g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs) [0] f[Ite][False][Ite](False, Cons(x, xs), y) -> xs [0] f[Ite][False][Ite](True, x', Cons(x, xs)) -> xs [0] f[Ite][False][Ite](False, x, y) -> Cons(Cons(Nil, Nil), y) [0] f[Ite][False][Ite](True, x, y) -> x [0] g[Ite][False][Ite](v0, v1, v2) -> null_g[Ite][False][Ite] [0] f[Ite][False][Ite](v0, v1, v2) -> null_f[Ite][False][Ite] [0] lt0(v0, v1) -> null_lt0 [0] g(v0, v1) -> null_g [0] f(v0, v1) -> null_f [0] The TRS has the following type information: lt0 :: Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f -> Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f -> False:True:null_lt0 Cons :: Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f -> Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f -> Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f g :: Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f -> Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f -> Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f Nil :: Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f f :: Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f -> Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f -> Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f False :: False:True:null_lt0 g[Ite][False][Ite] :: False:True:null_lt0 -> Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f -> Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f -> Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f f[Ite][False][Ite] :: False:True:null_lt0 -> Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f -> Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f -> Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f number42 :: Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f goal :: Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f -> Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f -> Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f True :: False:True:null_lt0 null_g[Ite][False][Ite] :: Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f null_f[Ite][False][Ite] :: Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f null_lt0 :: False:True:null_lt0 null_g :: Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f null_f :: Cons:Nil:null_g[Ite][False][Ite]:null_f[Ite][False][Ite]:null_g:null_f Rewrite Strategy: INNERMOST ---------------------------------------- (55) 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 False => 1 True => 2 null_g[Ite][False][Ite] => 0 null_f[Ite][False][Ite] => 0 null_lt0 => 0 null_g => 0 null_f => 0 ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> f(f[Ite][False][Ite](lt0(x, 1 + 0 + 0), x, 1 + x' + xs), f[Ite][False][Ite](lt0(x, 1 + 0 + 0), x, 1 + x' + xs)) :|: xs >= 0, z' = 1 + x' + xs, x >= 0, x' >= 0, z = x f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 f(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: x >= 0, z = x, z' = 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> x :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: xs >= 0, z' = 1 + x + xs, z'' = y, z = 1, x >= 0, y >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs f[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 f[Ite][False][Ite](z, z', z'') -{ 0 }-> 1 + (1 + 0 + 0) + y :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 g(z, z') -{ 1 }-> g[Ite][False][Ite](lt0(x, 1 + 0 + 0), x, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x >= 0, x' >= 0, z = x g(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 g(z, z') -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: x >= 0, z = x, z' = 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(x', xs) :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs g[Ite][False][Ite](z, z', z'') -{ 0 }-> g(xs, 1 + (1 + 0 + 0) + y) :|: xs >= 0, z' = 1 + x + xs, z'' = y, z = 1, x >= 0, y >= 0 g[Ite][False][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 goal(z, z') -{ 1 }-> 1 + f(x, y) + (1 + g(x, y) + 0) :|: x >= 0, y >= 0, z = x, z' = y lt0(z, z') -{ 1 }-> lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lt0(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0 lt0(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 number42 -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (57) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2)) g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2) f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)) f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) f(z0, Cons(z1, z2)) -> f(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))) number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil)) Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2 F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3 F[ITE][FALSE][ITE](False, z0, z1) -> c4 F[ITE][FALSE][ITE](True, z0, z1) -> c5 LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) LT0(z0, Nil) -> c7 G(z0, Nil) -> c8 G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Nil) -> c10 F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) NUMBER42 -> c13 GOAL(z0, z1) -> c14(F(z0, z1)) GOAL(z0, z1) -> c15(G(z0, z1)) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) LT0(z0, Nil) -> c7 G(z0, Nil) -> c8 G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Nil) -> c10 F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) NUMBER42 -> c13 GOAL(z0, z1) -> c14(F(z0, z1)) GOAL(z0, z1) -> c15(G(z0, z1)) K tuples:none Defined Rule Symbols: lt0_2, g_2, f_2, number42, goal_2, g[Ite][False][Ite]_3, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, F[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2, NUMBER42, GOAL_2 Compound Symbols: c_1, c1_1, c2, c3, c4, c5, c6_1, c7, c8, c9_2, c10, c11_3, c12_3, c13, c14_1, c15_1 ---------------------------------------- (59) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: GOAL(z0, z1) -> c15(G(z0, z1)) GOAL(z0, z1) -> c14(F(z0, z1)) Removed 7 trailing nodes: F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3 LT0(z0, Nil) -> c7 F(z0, Nil) -> c10 F[ITE][FALSE][ITE](False, z0, z1) -> c4 F[ITE][FALSE][ITE](True, z0, z1) -> c5 F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2 NUMBER42 -> c13 ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2)) g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2) f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)) f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) f(z0, Cons(z1, z2)) -> f(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))) number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil)) Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(z0, Nil) -> c8 G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(z0, Nil) -> c8 G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, g_2, f_2, number42, goal_2, g[Ite][False][Ite]_3, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2 Compound Symbols: c_1, c1_1, c6_1, c8, c9_2, c11_3, c12_3 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2)) g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2) f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)) f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) f(z0, Cons(z1, z2)) -> f(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))) number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil)) Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(z0, Nil) -> c8 G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(z0, Nil) -> c8 G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, g_2, f_2, number42, goal_2, g[Ite][False][Ite]_3, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2 Compound Symbols: c_1, c1_1, c6_1, c8, c9_2, c11_2, c12_2 ---------------------------------------- (63) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2)) g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2) g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)) f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) f(z0, Cons(z1, z2)) -> f(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))) number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(z0, Nil) -> c8 G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(z0, Nil) -> c8 G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2 Compound Symbols: c_1, c1_1, c6_1, c8, c9_2, c11_2, c12_2 ---------------------------------------- (65) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. G(z0, Nil) -> c8 We considered the (Usable) Rules:none And the Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(z0, Nil) -> c8 G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(Cons(x_1, x_2)) = 0 POL(F(x_1, x_2)) = 0 POL(False) = [1] POL(G(x_1, x_2)) = [1] POL(G[ITE][FALSE][ITE](x_1, x_2, x_3)) = [1] + x_3 POL(LT0(x_1, x_2)) = 0 POL(Nil) = 0 POL(True) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8) = 0 POL(c9(x_1, x_2)) = x_1 + x_2 POL(f[Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(lt0(x_1, x_2)) = [1] ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(z0, Nil) -> c8 G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) K tuples: G(z0, Nil) -> c8 Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2 Compound Symbols: c_1, c1_1, c6_1, c8, c9_2, c11_2, c12_2 ---------------------------------------- (67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) We considered the (Usable) Rules: lt0(z0, Nil) -> False lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) And the Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(z0, Nil) -> c8 G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(Cons(x_1, x_2)) = [1] + x_2 POL(F(x_1, x_2)) = 0 POL(False) = 0 POL(G(x_1, x_2)) = [1] + x_1 POL(G[ITE][FALSE][ITE](x_1, x_2, x_3)) = x_1 + x_2 POL(LT0(x_1, x_2)) = 0 POL(Nil) = 0 POL(True) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8) = 0 POL(c9(x_1, x_2)) = x_1 + x_2 POL(f[Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(lt0(x_1, x_2)) = 0 ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(z0, Nil) -> c8 G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) K tuples: G(z0, Nil) -> c8 G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2 Compound Symbols: c_1, c1_1, c6_1, c8, c9_2, c11_2, c12_2 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) by G(Cons(z0, z1), Cons(x1, x2)) -> c9(G[ITE][FALSE][ITE](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), LT0(Cons(z0, z1), Cons(Nil, Nil))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(z0, Nil) -> c8 F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) G(Cons(z0, z1), Cons(x1, x2)) -> c9(G[ITE][FALSE][ITE](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), LT0(Cons(z0, z1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) K tuples: G(z0, Nil) -> c8 G(z0, Cons(z1, z2)) -> c9(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2 Compound Symbols: c_1, c1_1, c6_1, c8, c11_2, c12_2, c9_2 ---------------------------------------- (71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: G(z0, Nil) -> c8 ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) G(Cons(z0, z1), Cons(x1, x2)) -> c9(G[ITE][FALSE][ITE](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), LT0(Cons(z0, z1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, F_2, G_2 Compound Symbols: c_1, c1_1, c6_1, c11_2, c12_2, c9_2 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(z0, Cons(z1, z2)) -> c11(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) G(Cons(z0, z1), Cons(x1, x2)) -> c9(G[ITE][FALSE][ITE](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, F_2, G_2 Compound Symbols: c_1, c1_1, c6_1, c12_2, c9_2, c11_2 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(z0, Cons(z1, z2)) -> c12(F(f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))), LT0(z0, Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(Cons(z0, z1), Cons(x1, x2)) -> c9(G[ITE][FALSE][ITE](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2 Compound Symbols: c_1, c1_1, c6_1, c9_2, c11_2, c12_2 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(Cons(z0, z1), Cons(x1, x2)) -> c9(G[ITE][FALSE][ITE](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), LT0(Cons(z0, z1), Cons(Nil, Nil))) by G(Cons(x0, z0), Cons(x2, x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x0, z0), Cons(x2, x3)), LT0(Cons(x0, z0), Cons(Nil, Nil))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G(Cons(x0, z0), Cons(x2, x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x0, z0), Cons(x2, x3)), LT0(Cons(x0, z0), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, F_2, G_2 Compound Symbols: c_1, c1_1, c6_1, c11_2, c12_2, c9_2 ---------------------------------------- (79) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G(Cons(x0, z0), Cons(x2, x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x0, z0), Cons(x2, x3)), LT0(Cons(x0, z0), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, F_2, G_2 Compound Symbols: c_1, c6_1, c11_2, c12_2, c9_2 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G(Cons(x0, z0), Cons(x2, x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x0, z0), Cons(x2, x3)), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, F_2, G_2 Compound Symbols: c_1, c6_1, c11_2, c12_2, c9_2 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(z0, z1), Cons(x1, x2)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G(Cons(x0, z0), Cons(x2, x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x0, z0), Cons(x2, x3)), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, F_2, G_2 Compound Symbols: c_1, c6_1, c12_2, c9_2, c11_2 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G(Cons(x0, z0), Cons(x2, x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x0, z0), Cons(x2, x3)), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, F_2, G_2 Compound Symbols: c_1, c6_1, c12_2, c9_2, c11_2 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(z0, z1), Cons(x1, x2)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x1, x2)), f[Ite][False][Ite](lt0(Cons(z0, z1), Cons(Nil, Nil)), Cons(z0, z1), Cons(x1, x2))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(Cons(x0, z0), Cons(x2, x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x0, z0), Cons(x2, x3)), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2 Compound Symbols: c_1, c6_1, c9_2, c11_2, c12_2 ---------------------------------------- (89) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(Cons(x0, z0), Cons(x2, x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x0, z0), Cons(x2, x3)), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2 Compound Symbols: c_1, c6_1, c9_2, c11_2, c12_2 ---------------------------------------- (91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) by G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(Cons(x0, z0), Cons(x2, x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x0, z0), Cons(x2, x3)), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G_2, F_2, G[ITE][FALSE][ITE]_3 Compound Symbols: c6_1, c9_2, c11_2, c12_2, c_1 ---------------------------------------- (93) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(Cons(x0, z0), Cons(x2, x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x0, z0), Cons(x2, x3)), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G_2, F_2, G[ITE][FALSE][ITE]_3 Compound Symbols: c6_1, c9_2, c11_2, c12_2, c_1 ---------------------------------------- (95) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(Cons(x0, z0), Cons(x2, x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x0, z0), Cons(x2, x3)), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G_2, F_2, G[ITE][FALSE][ITE]_3 Compound Symbols: c6_1, c9_2, c11_2, c12_2, c_1 ---------------------------------------- (97) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(Cons(x0, z0), Cons(x2, x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x0, z0), Cons(x2, x3)), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G_2, F_2, G[ITE][FALSE][ITE]_3 Compound Symbols: c6_1, c9_2, c12_2, c11_2, c_1 ---------------------------------------- (99) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(Cons(x0, z0), Cons(x2, x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x0, z0), Cons(x2, x3)), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G_2, F_2, G[ITE][FALSE][ITE]_3 Compound Symbols: c6_1, c9_2, c12_2, c11_2, c_1 ---------------------------------------- (101) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace G(Cons(x0, z0), Cons(x2, x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x0, z0), Cons(x2, x3)), LT0(Cons(x0, z0), Cons(Nil, Nil))) by G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, F_2, G[ITE][FALSE][ITE]_3, G_2 Compound Symbols: c6_1, c12_2, c11_2, c_1, c9_2 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, F_2, G[ITE][FALSE][ITE]_3, G_2 Compound Symbols: c6_1, c12_2, c11_2, c_1, c9_2 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, F_2, G[ITE][FALSE][ITE]_3, G_2 Compound Symbols: c6_1, c12_2, c11_2, c_1, c9_2 ---------------------------------------- (107) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](lt0(Cons(x0, z0), Cons(Nil, Nil)), Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) lt0(z0, Nil) -> False f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: lt0_2, f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, F_2, G[ITE][FALSE][ITE]_3, G_2 Compound Symbols: c6_1, c11_2, c_1, c12_2, c9_2 ---------------------------------------- (109) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, F_2, G[ITE][FALSE][ITE]_3, G_2 Compound Symbols: c6_1, c11_2, c_1, c12_2, c9_2 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, F_2, G_2 Compound Symbols: c6_1, c_1, c11_2, c12_2, c9_2 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(z0, z1), Cons(z2, z3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, F_2, G_2 Compound Symbols: c6_1, c_1, c12_2, c9_2, c11_2 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c12_2, c11_2 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(z0, z1), Cons(z2, z3)) -> c12(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(z2, z3)), f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(z2, z3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_2, c12_2 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_2, c12_2 ---------------------------------------- (121) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) by F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_2, c12_2 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples:none Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_2, c12_2, c11_1 ---------------------------------------- (125) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) We considered the (Usable) Rules:none And the Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(Cons(x_1, x_2)) = 0 POL(F(x_1, x_2)) = [1] POL(False) = 0 POL(G(x_1, x_2)) = x_1 POL(G[ITE][FALSE][ITE](x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(LT0(x_1, x_2)) = 0 POL(Nil) = 0 POL(True) = 0 POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(f[Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_1 POL(lt0(x_1, x_2)) = [1] + x_1 + x_2 ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_2, c12_2, c11_1 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_2, c12_2, c11_1 ---------------------------------------- (129) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(z0, f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_2, c12_2, c11_1 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) by F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c12_2, c11_2, c11_1 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c12_2, c11_2, c11_1 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) by F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c12_2, c11_2, c11_1 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c12_2, c11_2, c11_1, c12_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. F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) We considered the (Usable) Rules:none And the Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(Cons(x_1, x_2)) = 0 POL(F(x_1, x_2)) = [1] POL(False) = [1] POL(G(x_1, x_2)) = x_1 POL(G[ITE][FALSE][ITE](x_1, x_2, x_3)) = x_2 + x_3 POL(LT0(x_1, x_2)) = 0 POL(Nil) = 0 POL(True) = 0 POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(f[Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(lt0(x_1, x_2)) = [1] + x_1 + x_2 ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c12_2, c11_2, c11_1, c12_1 ---------------------------------------- (141) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c12_2, c11_2, c11_1, c12_1 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](lt0(z1, Nil), Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(z0, f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c12_2, c11_2, c11_1, c12_1 ---------------------------------------- (145) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](lt0(x1, Nil), Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) by F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3, lt0_2 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (147) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> z2 f[Ite][False][Ite](True, z0, z1) -> z0 lt0(z0, Nil) -> False ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (149) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (151) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (153) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(x0, z0), Cons(x2, x3)) -> c11(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (155) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (157) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (159) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (161) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_1, c12_2, c12_1, c11_2 ---------------------------------------- (163) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_1, c12_2, c12_1, c11_2 ---------------------------------------- (165) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), z0), LT0(Cons(x0, z0), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_1, c12_2, c12_1, c11_2 ---------------------------------------- (167) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_1, c12_2, c12_1, c11_2 ---------------------------------------- (169) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(x0, z0), Cons(x2, x3)) -> c12(F(f[Ite][False][Ite](False, Cons(x0, z0), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, z0), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_1, c12_2, c12_1, c11_2 ---------------------------------------- (171) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_1, c12_2, c12_1, c11_2 ---------------------------------------- (173) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, f[Ite][False][Ite](False, Cons(z0, z1), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_1, c12_2, c12_1, c11_2 ---------------------------------------- (175) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_1, c12_2, c12_1, c11_2 ---------------------------------------- (177) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), f[Ite][False][Ite](False, Cons(x0, x1), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) by F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_1, c12_1, c11_2, c12_2 ---------------------------------------- (179) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols: f[Ite][False][Ite]_3 Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_1, c12_1, c11_2, c12_2 ---------------------------------------- (181) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f[Ite][False][Ite](False, Cons(z0, z1), z2) -> z1 f[Ite][False][Ite](False, z0, z1) -> Cons(Cons(Nil, Nil), z1) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols:none Defined Pair Symbols: LT0_2, G[ITE][FALSE][ITE]_3, G_2, F_2 Compound Symbols: c6_1, c_1, c9_2, c11_1, c12_1, c11_2, c12_2 ---------------------------------------- (183) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace G[ITE][FALSE][ITE](False, Cons(z0, Cons(y0, y1)), z2) -> c(G(Cons(y0, y1), Cons(Cons(Nil, Nil), z2))) by G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), x2)) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2)))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), x2)) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2)))) S tuples: LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols:none Defined Pair Symbols: LT0_2, G_2, F_2, G[ITE][FALSE][ITE]_3 Compound Symbols: c6_1, c9_2, c11_1, c12_1, c11_2, c12_2, c_1 ---------------------------------------- (185) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LT0(Cons(z0, z1), Cons(z2, z3)) -> c6(LT0(z1, z3)) by LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), x2)) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2)))) LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) S tuples: F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) K tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) Defined Rule Symbols:none Defined Pair Symbols: G_2, F_2, G[ITE][FALSE][ITE]_3, LT0_2 Compound Symbols: c9_2, c11_1, c12_1, c11_2, c12_2, c_1, c6_1 ---------------------------------------- (187) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: F(Cons(x0, x1), Cons(x2, x3)) -> c12(LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(LT0(Cons(x0, x1), Cons(Nil, Nil))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3)), LT0(Cons(x1, x2), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), x2)) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2)))) LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) S tuples: F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1), LT0(Cons(x0, x1), Cons(Nil, Nil))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(z0, z1), Cons(Nil, Nil))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))), LT0(Cons(x0, x1), Cons(Nil, Nil))) LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: G_2, F_2, G[ITE][FALSE][ITE]_3, LT0_2 Compound Symbols: c9_2, c11_2, c12_2, c_1, c6_1 ---------------------------------------- (189) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing tuple parts ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), x2)) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2)))) LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) S tuples: LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2 Compound Symbols: c_1, c6_1, c9_1, c11_1, c12_1 ---------------------------------------- (191) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) We considered the (Usable) Rules:none And the Tuples: G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), x2)) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2)))) LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(Cons(x_1, x_2)) = [1] + x_2 POL(F(x_1, x_2)) = 0 POL(False) = 0 POL(G(x_1, x_2)) = 0 POL(G[ITE][FALSE][ITE](x_1, x_2, x_3)) = [3]x_1 POL(LT0(x_1, x_2)) = [2]x_1 POL(Nil) = 0 POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c9(x_1)) = x_1 ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), x2)) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2)))) LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) S tuples: F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) K tuples: LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) Defined Rule Symbols:none Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2 Compound Symbols: c_1, c6_1, c9_1, c11_1, c12_1 ---------------------------------------- (193) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace G(Cons(x1, x2), Cons(Cons(Nil, Nil), x3)) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), x3))) by G(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3))) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3)))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), x2)) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2)))) LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3))) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3)))) S tuples: F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) K tuples: LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) Defined Rule Symbols:none Defined Pair Symbols: G[ITE][FALSE][ITE]_3, LT0_2, F_2, G_2 Compound Symbols: c_1, c6_1, c11_1, c12_1, c9_1 ---------------------------------------- (195) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), x2)) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2)))) by G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3))) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3)))) G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))))) S tuples: F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) K tuples: LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) Defined Rule Symbols:none Defined Pair Symbols: LT0_2, F_2, G_2, G[ITE][FALSE][ITE]_3 Compound Symbols: c6_1, c11_1, c12_1, c9_1, c_1 ---------------------------------------- (197) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace G(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3))) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3)))) by G(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3))))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3))))) S tuples: F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) K tuples: LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) Defined Rule Symbols:none Defined Pair Symbols: LT0_2, F_2, G[ITE][FALSE][ITE]_3, G_2 Compound Symbols: c6_1, c11_1, c12_1, c_1, c9_1 ---------------------------------------- (199) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LT0(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c6(LT0(Cons(y0, y1), Cons(y2, y3))) by LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3))))) LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) S tuples: F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: F_2, G[ITE][FALSE][ITE]_3, G_2, LT0_2 Compound Symbols: c11_1, c12_1, c_1, c9_1, c6_1 ---------------------------------------- (201) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, z1)) by F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(y0, y1))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3))))) LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(y0, y1))) S tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(y0, y1))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: F_2, G[ITE][FALSE][ITE]_3, G_2, LT0_2 Compound Symbols: c11_1, c12_1, c_1, c9_1, c6_1 ---------------------------------------- (203) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) by F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3))))) LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) S tuples: F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: F_2, G[ITE][FALSE][ITE]_3, G_2, LT0_2 Compound Symbols: c11_1, c12_1, c_1, c9_1, c6_1 ---------------------------------------- (205) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(Cons(z0, z1), Cons(x2, x3)) -> c11(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) by F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3))))) LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) S tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: F_2, G[ITE][FALSE][ITE]_3, G_2, LT0_2 Compound Symbols: c11_1, c12_1, c_1, c9_1, c6_1 ---------------------------------------- (207) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(y0, y1))) by F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3))))) LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) S tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: F_2, G[ITE][FALSE][ITE]_3, G_2, LT0_2 Compound Symbols: c11_1, c12_1, c_1, c9_1, c6_1 ---------------------------------------- (209) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, z1)) by F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3))))) LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) S tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: F_2, G[ITE][FALSE][ITE]_3, G_2, LT0_2 Compound Symbols: c11_1, c12_1, c_1, c9_1, c6_1 ---------------------------------------- (211) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), x1)) by F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3))))) LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) S tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: F_2, G[ITE][FALSE][ITE]_3, G_2, LT0_2 Compound Symbols: c11_1, c12_1, c_1, c9_1, c6_1 ---------------------------------------- (213) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(Cons(z0, z1), Cons(x2, x3)) -> c12(F(z1, Cons(Cons(Nil, Nil), Cons(x2, x3)))) by F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))))) G(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3))))) LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) S tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: F_2, G[ITE][FALSE][ITE]_3, G_2, LT0_2 Compound Symbols: c11_1, c12_1, c_1, c9_1, c6_1 ---------------------------------------- (215) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace G(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x3))))) by G(Cons(z0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z2)))) -> c9(G[ITE][FALSE][ITE](False, Cons(z0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z2))))) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))))) LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) G(Cons(z0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z2)))) -> c9(G[ITE][FALSE][ITE](False, Cons(z0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z2))))) S tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: F_2, G[ITE][FALSE][ITE]_3, LT0_2, G_2 Compound Symbols: c11_1, c12_1, c_1, c6_1, c9_1 ---------------------------------------- (217) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace G[ITE][FALSE][ITE](False, Cons(x0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))) -> c(G(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x2))))) by G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))) -> c(G(Cons(z1, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))))) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) G(Cons(z0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z2)))) -> c9(G[ITE][FALSE][ITE](False, Cons(z0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z2))))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))) -> c(G(Cons(z1, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))))) S tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: F_2, LT0_2, G_2, G[ITE][FALSE][ITE]_3 Compound Symbols: c11_1, c12_1, c6_1, c9_1, c_1 ---------------------------------------- (219) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace G(Cons(z0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z2)))) -> c9(G[ITE][FALSE][ITE](False, Cons(z0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z2))))) by G(Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))))) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))) -> c(G(Cons(z1, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))))) G(Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))))) S tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: F_2, LT0_2, G[ITE][FALSE][ITE]_3, G_2 Compound Symbols: c11_1, c12_1, c6_1, c_1, c9_1 ---------------------------------------- (221) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))) -> c(G(Cons(z1, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))))) by G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(z2, Cons(y2, y3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z4))) -> c(G(Cons(z1, Cons(z2, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z4))))) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) G(Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(z2, Cons(y2, y3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z4))) -> c(G(Cons(z1, Cons(z2, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z4))))) S tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: F_2, LT0_2, G_2, G[ITE][FALSE][ITE]_3 Compound Symbols: c11_1, c12_1, c6_1, c9_1, c_1 ---------------------------------------- (223) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(Cons(x0, x1), Cons(x2, x3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) by F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) G(Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(z2, Cons(y2, y3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z4))) -> c(G(Cons(z1, Cons(z2, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z4))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) S tuples: F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: F_2, LT0_2, G_2, G[ITE][FALSE][ITE]_3 Compound Symbols: c12_1, c6_1, c11_1, c9_1, c_1 ---------------------------------------- (225) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(Cons(x0, x1), Cons(x2, x3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3)))) by F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) G(Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(z2, Cons(y2, y3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z4))) -> c(G(Cons(z1, Cons(z2, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z4))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) S tuples: F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: LT0_2, F_2, G_2, G[ITE][FALSE][ITE]_3 Compound Symbols: c6_1, c11_1, c12_1, c9_1, c_1 ---------------------------------------- (227) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c11(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) by F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, x4))) F(Cons(x1, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(z1, z2))) F(Cons(x0, Cons(z1, z2)), Cons(x0, Cons(z1, z2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, z2))), Cons(z1, z2))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(x2, Cons(x3, x4)))) F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(x1, x2))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(x3, x4))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(x4, x5))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, Cons(x4, Cons(x5, x6))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x3, x4)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(x2, Cons(x3, x4)))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(x2, Cons(x3, Cons(x4, x5))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(x2, x3))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(x0, Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(x0, Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) G(Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(z2, Cons(y2, y3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z4))) -> c(G(Cons(z1, Cons(z2, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z4))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, x4))) F(Cons(x1, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(z1, z2))) F(Cons(x0, Cons(z1, z2)), Cons(x0, Cons(z1, z2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, z2))), Cons(z1, z2))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(x2, Cons(x3, x4)))) F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(x1, x2))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(x3, x4))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(x4, x5))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, Cons(x4, Cons(x5, x6))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x3, x4)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(x2, Cons(x3, x4)))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(x2, Cons(x3, Cons(x4, x5))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(x2, x3))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(x0, Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(x0, Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) S tuples: F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, x4))) F(Cons(x1, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(z1, z2))) F(Cons(x0, Cons(z1, z2)), Cons(x0, Cons(z1, z2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, z2))), Cons(z1, z2))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(x2, Cons(x3, x4)))) F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(x1, x2))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(x3, x4))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(x4, x5))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, Cons(x4, Cons(x5, x6))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x3, x4)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(x2, Cons(x3, x4)))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(x2, Cons(x3, Cons(x4, x5))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(x2, x3))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(x0, Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(x0, Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: LT0_2, F_2, G_2, G[ITE][FALSE][ITE]_3 Compound Symbols: c6_1, c11_1, c12_1, c9_1, c_1 ---------------------------------------- (229) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c11(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) by F(Cons(x1, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x0, Cons(z1, z2)), Cons(x0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, z2))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(x2, Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(x3, x4), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(x3, Cons(x4, x5)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(x3, x4), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(x4, x5), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(x3, Cons(x4, Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(x2, Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(x2, Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(x2, x3), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(x0, Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(x0, Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(x1, x2)) -> c11(F(Cons(x0, Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(x0, Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x0, Cons(x1, Cons(x2, x3)))) -> c11(F(Cons(x4, x5), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x5, x6)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(x0, x1))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(x0, Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(x0, Cons(x1, Cons(x2, x3)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))))) ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) G(Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(z2, Cons(y2, y3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z4))) -> c(G(Cons(z1, Cons(z2, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z4))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, x4))) F(Cons(x1, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(z1, z2))) F(Cons(x0, Cons(z1, z2)), Cons(x0, Cons(z1, z2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, z2))), Cons(z1, z2))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(x2, Cons(x3, x4)))) F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(x1, x2))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(x3, x4))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(x4, x5))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, Cons(x4, Cons(x5, x6))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x3, x4)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(x2, Cons(x3, x4)))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(x2, Cons(x3, Cons(x4, x5))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(x2, x3))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(x0, Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(x0, Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(x1, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x0, Cons(z1, z2)), Cons(x0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, z2))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(x2, Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(x3, x4), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(x3, Cons(x4, x5)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(x3, x4), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(x4, x5), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(x3, Cons(x4, Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(x2, Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(x2, Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(x2, x3), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(x0, Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(x0, Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(x1, x2)) -> c11(F(Cons(x0, Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(x0, Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x0, Cons(x1, Cons(x2, x3)))) -> c11(F(Cons(x4, x5), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x5, x6)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(x0, x1))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(x0, Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(x0, Cons(x1, Cons(x2, x3)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))))) S tuples: F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, x4))) F(Cons(x1, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(z1, z2))) F(Cons(x0, Cons(z1, z2)), Cons(x0, Cons(z1, z2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, z2))), Cons(z1, z2))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(x2, Cons(x3, x4)))) F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(x1, x2))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(x3, x4))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(x4, x5))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, Cons(x4, Cons(x5, x6))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x3, x4)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(x2, Cons(x3, x4)))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(x2, Cons(x3, Cons(x4, x5))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(x2, x3))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(x0, Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(x0, Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(x1, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x0, Cons(z1, z2)), Cons(x0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, z2))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(x2, Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(x3, x4), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(x3, Cons(x4, x5)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(x3, x4), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(x4, x5), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(x3, Cons(x4, Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(x2, Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(x2, Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(x2, x3), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(x0, Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(x0, Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(x1, x2)) -> c11(F(Cons(x0, Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(x0, Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x0, Cons(x1, Cons(x2, x3)))) -> c11(F(Cons(x4, x5), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x5, x6)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(x0, x1))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(x0, Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(x0, Cons(x1, Cons(x2, x3)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: LT0_2, F_2, G_2, G[ITE][FALSE][ITE]_3 Compound Symbols: c6_1, c11_1, c12_1, c9_1, c_1 ---------------------------------------- (231) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c11(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) by F(Cons(x1, Cons(x2, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(x2, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, Cons(z1, Cons(z2, z3))), Cons(x0, Cons(z1, Cons(z2, z3)))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, Cons(z2, z3)))))) F(Cons(Cons(Nil, Nil), Cons(x1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(z2, z3)))) -> c11(F(Cons(x1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, Cons(z2, z3)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(z2, z3))), Cons(x1, x2)) -> c11(F(Cons(x3, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(x3, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, Cons(z2, z3))), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(x4, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(x3, Cons(x4, Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(x2, Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(x2, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(x0, Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(x1, x2)) -> c11(F(Cons(x0, Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(x0, Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(Cons(Nil, Nil), Cons(x4, Cons(z2, z3))), Cons(x0, Cons(x1, Cons(x2, x3)))) -> c11(F(Cons(x4, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x5, x6)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(x0, x1))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(x0, Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(x0, Cons(x1, Cons(x2, x3)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))))) F(Cons(x1, Cons(z1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4)))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4)))))) F(Cons(x1, Cons(z1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(z1, Cons(z2, z3)))))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(z1, Cons(z2, z3)))))))) F(Cons(x0, Cons(z1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, Cons(z2, z3)))))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, Cons(z2, z3)))))))) F(Cons(Cons(Nil, Nil), Cons(x1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4)))) -> c11(F(Cons(x1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4)))))) F(Cons(x1, Cons(z1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(z2, z3)))))) -> c11(F(Cons(x0, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(z2, z3)))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) -> c11(F(Cons(x0, Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))))))) F(Cons(x0, Cons(x1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(z2, z3)))) -> c11(F(Cons(x1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, Cons(z2, z3)))))) F(Cons(x0, Cons(z1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(x0, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(x0, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, Cons(x4, x5))))))) ---------------------------------------- (232) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) G(Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3)))) -> c9(G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z3))))) G[ITE][FALSE][ITE](False, Cons(z0, Cons(z1, Cons(z2, Cons(y2, y3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z4))) -> c(G(Cons(z1, Cons(z2, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), z4))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, x4))) F(Cons(x1, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(z1, z2))) F(Cons(x0, Cons(z1, z2)), Cons(x0, Cons(z1, z2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, z2))), Cons(z1, z2))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(x2, Cons(x3, x4)))) F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(x1, x2))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(x3, x4))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(x4, x5))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, Cons(x4, Cons(x5, x6))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x3, x4)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(x2, Cons(x3, x4)))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(x2, Cons(x3, Cons(x4, x5))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(x2, x3))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(x0, Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(x0, Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(x1, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x0, Cons(z1, z2)), Cons(x0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, z2))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(x2, Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(x3, x4), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(x3, Cons(x4, x5)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(x3, x4), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(x4, x5), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(x3, Cons(x4, Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(x2, Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(x2, Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(x2, x3), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(x0, Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(x0, Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(x1, x2)) -> c11(F(Cons(x0, Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(x0, Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x0, Cons(x1, Cons(x2, x3)))) -> c11(F(Cons(x4, x5), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x5, x6)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(x0, x1))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(x0, Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(x0, Cons(x1, Cons(x2, x3)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(x3, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, Cons(z2, z3))), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(x4, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(x2, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(x4, Cons(z2, z3))), Cons(x0, Cons(x1, Cons(x2, x3)))) -> c11(F(Cons(x4, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(x1, Cons(z1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4)))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4)))))) F(Cons(x1, Cons(z1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(z1, Cons(z2, z3)))))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(z1, Cons(z2, z3)))))))) F(Cons(x0, Cons(z1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, Cons(z2, z3)))))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, Cons(z2, z3)))))))) F(Cons(Cons(Nil, Nil), Cons(x1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4)))) -> c11(F(Cons(x1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4)))))) F(Cons(x1, Cons(z1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(z2, z3)))))) -> c11(F(Cons(x0, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(z2, z3)))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) -> c11(F(Cons(x0, Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))))))) F(Cons(x0, Cons(x1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(z2, z3)))) -> c11(F(Cons(x1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, Cons(z2, z3)))))) F(Cons(x0, Cons(z1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(x0, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(x0, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, Cons(x4, x5))))))) S tuples: F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(z0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(z3, z4)) -> c11(F(Cons(x2, x3), Cons(x2, x3))) F(Cons(z0, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(z1, z2), Cons(z1, z2))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(y0, y1))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(y0, Cons(y1, y2)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(y0, Cons(y1, Cons(y2, y3))))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(y0, y1)))) F(Cons(z0, Cons(y2, y3)), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y2, y3))) F(Cons(z0, Cons(y4, y5)), Cons(z2, Cons(y2, y3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, y3))), Cons(y4, y5))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y2, y3))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(y2, y3)))) F(Cons(z0, Cons(y3, Cons(y4, y5))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, z3)), Cons(y3, Cons(y4, y5)))) F(Cons(z0, Cons(y5, y6)), Cons(z2, Cons(y2, Cons(y3, y4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(z2, Cons(y2, Cons(y3, y4)))), Cons(y5, y6))) F(Cons(z0, Cons(y3, y4)), Cons(Cons(Nil, Nil), Cons(y1, y2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(y3, y4))) F(Cons(z0, Cons(y0, y1)), Cons(z2, z3)) -> c12(F(Cons(y0, y1), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, y3)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, y3))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(Cons(Nil, Nil), Cons(y0, y1))), Cons(z2, z3)) -> c12(F(Cons(Cons(Nil, Nil), Cons(y0, y1)), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4))))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(y1, Cons(y2, Cons(y3, y4)))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2)))), Cons(z2, z3)) -> c12(F(Cons(y0, Cons(Cons(Nil, Nil), Cons(y1, y2))), Cons(Cons(Nil, Nil), Cons(z2, z3)))) F(Cons(z0, Cons(y0, Cons(y1, y2))), Cons(z2, Cons(y5, y6))) -> c12(F(Cons(y0, Cons(y1, y2)), Cons(Cons(Nil, Nil), Cons(z2, Cons(y5, y6))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x0, x1), Cons(x0, x1)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(x1, Cons(x2, x3)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c12(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c12(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, x4))) F(Cons(x1, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(z1, z2))) F(Cons(x0, Cons(z1, z2)), Cons(x0, Cons(z1, z2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, z2))), Cons(z1, z2))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))), Cons(x2, Cons(x3, x4)))) F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(x1, x2))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, Cons(x4, x5)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))), Cons(x3, x4))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))), Cons(x4, x5))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(x3, Cons(x4, Cons(x5, x6))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x3, x4)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(x2, Cons(x3, x4)))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))), Cons(x2, Cons(x3, Cons(x4, x5))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(x2, x3)))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))), Cons(x2, x3))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(x0, Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(x0, Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(x1, Cons(z1, z2)), Cons(Cons(Nil, Nil), Cons(x3, x4))) -> c11(F(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))))) F(Cons(x0, Cons(z1, z2)), Cons(x0, Cons(z1, z2))) -> c11(F(Cons(z1, z2), Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, z2))))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(x2, Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(x1, x2), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(x1, x2)) -> c11(F(Cons(x3, x4), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(x1, x2)) -> c11(F(Cons(x3, Cons(x4, x5)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(x3, x4), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(x4, x5), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, Cons(x5, x6)))), Cons(x1, x2)) -> c11(F(Cons(x3, Cons(x4, Cons(x5, x6))), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(x1, Cons(x2, Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x5, x6))) -> c11(F(Cons(x2, Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))))) F(Cons(x1, Cons(x2, Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(x6, x7))) -> c11(F(Cons(x2, Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x6, x7))))) F(Cons(x1, Cons(Cons(Nil, Nil), Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x4, x5))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))))) F(Cons(x1, Cons(x2, x3)), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(x2, x3), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(x0, Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(x0, Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(x1, x2)) -> c11(F(Cons(x0, Cons(x1, x2)), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(x0, Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(x0, x1)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(x0, x1)))) F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(x0, Cons(x1, Cons(x2, x3)))) -> c11(F(Cons(x4, x5), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, x5))), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x4, x5)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x5, x6))), Cons(x1, Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x5, x6)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4))), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, x4)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5)))), Cons(x1, x2)) -> c11(F(Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))), Cons(Cons(Nil, Nil), Cons(x1, x2)))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(x0, x1))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(x0, Cons(x1, x2))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(x0, Cons(x1, Cons(x2, x3)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))))) F(Cons(Cons(Nil, Nil), Cons(x3, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x1, x2))) -> c11(F(Cons(x3, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, x2))))) F(Cons(Cons(Nil, Nil), Cons(x4, Cons(z2, z3))), Cons(x1, Cons(x2, x3))) -> c11(F(Cons(x4, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(x1, Cons(x2, x3))))) F(Cons(x1, Cons(x2, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))) -> c11(F(Cons(x2, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x4, Cons(x5, x6)))))) F(Cons(Cons(Nil, Nil), Cons(x4, Cons(z2, z3))), Cons(x0, Cons(x1, Cons(x2, x3)))) -> c11(F(Cons(x4, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))))) F(Cons(x1, Cons(z1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4)))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4)))))) F(Cons(x1, Cons(z1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(z1, Cons(z2, z3)))))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(z1, Cons(z2, z3)))))))) F(Cons(x0, Cons(z1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, Cons(z2, z3)))))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(z1, Cons(z2, z3)))))))) F(Cons(Cons(Nil, Nil), Cons(x1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4)))) -> c11(F(Cons(x1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, x4)))))) F(Cons(x1, Cons(z1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(z2, z3)))))) -> c11(F(Cons(x0, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(z2, z3)))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, x1)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, x1)))))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))) -> c11(F(Cons(x0, Cons(x1, Cons(x2, x3))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, Cons(x2, x3))))))))) F(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2)))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))))) -> c11(F(Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x0, Cons(x1, x2))))))))) F(Cons(x0, Cons(x1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x1, Cons(z2, z3)))) -> c11(F(Cons(x1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x1, Cons(z2, z3)))))) F(Cons(x0, Cons(z1, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(z1, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, Cons(x4, x5))))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, x4)))) -> c11(F(Cons(x0, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, x4)))))) F(Cons(Cons(Nil, Nil), Cons(x0, Cons(z2, z3))), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, Cons(x4, x5))))) -> c11(F(Cons(x0, Cons(z2, z3)), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(x2, Cons(x3, Cons(x4, x5))))))) K tuples: LT0(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c6(LT0(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) Defined Rule Symbols:none Defined Pair Symbols: LT0_2, F_2, G_2, G[ITE][FALSE][ITE]_3 Compound Symbols: c6_1, c11_1, c12_1, c9_1, c_1