KILLED proof of input_abUxujihlW.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (6) CdtProblem (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxRelTRS (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (10) TRS for Loop Detection (11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxRelTRS (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (14) typed CpxTrs (15) OrderProof [LOWER BOUND(ID), 0 ms] (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 2041 ms] (18) typed CpxTrs (19) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (20) CdtProblem (21) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CdtProblem (23) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxRelTRS (25) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (26) CpxTRS (27) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CpxWeightedTrs (29) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CpxTypedWeightedTrs (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) CompletionProof [UPPER BOUND(ID), 0 ms] (36) CpxTypedWeightedCompleteTrs (37) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CpxTypedWeightedCompleteTrs (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CpxRNTS (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms] (90) CdtProblem (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 18 ms] (110) CdtProblem (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 10 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 30 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 12 ms] (158) CdtProblem (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 108 ms] (160) CdtProblem (161) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (162) CpxWeightedTrs (163) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CpxTypedWeightedTrs (165) CompletionProof [UPPER BOUND(ID), 0 ms] (166) CpxTypedWeightedCompleteTrs (167) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CpxTypedWeightedCompleteTrs (169) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (170) CpxRNTS (171) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CpxRNTS (173) CompletionProof [UPPER BOUND(ID), 0 ms] (174) CpxTypedWeightedCompleteTrs (175) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (176) CpxRNTS ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, 0, 0) -> s(x) f(0, y, 0) -> s(y) f(0, 0, z) -> s(z) f(s(0), y, z) -> f(0, s(y), s(z)) f(s(x), s(y), 0) -> f(x, y, s(0)) f(s(x), 0, s(z)) -> f(x, s(0), z) f(0, s(0), s(0)) -> s(s(0)) f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) f(0, s(s(y)), s(0)) -> f(0, y, s(0)) f(0, s(0), s(s(z))) -> f(0, s(0), z) f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, 0', 0') -> s(x) f(0', y, 0') -> s(y) f(0', 0', z) -> s(z) f(s(0'), y, z) -> f(0', s(y), s(z)) f(s(x), s(y), 0') -> f(x, y, s(0')) f(s(x), 0', s(z)) -> f(x, s(0'), z) f(0', s(0'), s(0')) -> s(s(0')) f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) f(0', s(s(y)), s(0')) -> f(0', y, s(0')) f(0', s(0'), s(s(z))) -> f(0', s(0'), z) f(0', s(s(y)), s(s(z))) -> f(0', y, f(0', s(s(y)), s(z))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, 0, 0) -> s(x) f(0, y, 0) -> s(y) f(0, 0, z) -> s(z) f(s(0), y, z) -> f(0, s(y), s(z)) f(s(x), s(y), 0) -> f(x, y, s(0)) f(s(x), 0, s(z)) -> f(x, s(0), z) f(0, s(0), s(0)) -> s(s(0)) f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) f(0, s(s(y)), s(0)) -> f(0, y, s(0)) f(0, s(0), s(s(z))) -> f(0, s(0), z) f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(z0, 0, 0) -> c F(0, z0, 0) -> c1 F(0, 0, z0) -> c2 F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(0), s(0)) -> c6 F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) S tuples: F(z0, 0, 0) -> c F(0, z0, 0) -> c1 F(0, 0, z0) -> c2 F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(0), s(0)) -> c6 F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c, c1, c2, c3_1, c4_1, c5_1, c6, c7_2, c8_1, c9_1, c10_2 ---------------------------------------- (7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: F(z0, 0, 0) -> c F(0, z0, 0) -> c1 F(0, 0, z0) -> c2 F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(0), s(0)) -> c6 F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) The (relative) TRS S consists of the following rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Rewrite Strategy: INNERMOST ---------------------------------------- (9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: F(z0, 0, 0) -> c F(0, z0, 0) -> c1 F(0, 0, z0) -> c2 F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(0), s(0)) -> c6 F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) The (relative) TRS S consists of the following rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Rewrite Strategy: INNERMOST ---------------------------------------- (11) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (12) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: F(z0, 0', 0') -> c F(0', z0, 0') -> c1 F(0', 0', z0) -> c2 F(s(0'), z0, z1) -> c3(F(0', s(z0), s(z1))) F(s(z0), s(z1), 0') -> c4(F(z0, z1, s(0'))) F(s(z0), 0', s(z1)) -> c5(F(z0, s(0'), z1)) F(0', s(0'), s(0')) -> c6 F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0', s(s(z0)), s(0')) -> c8(F(0', z0, s(0'))) F(0', s(0'), s(s(z0))) -> c9(F(0', s(0'), z0)) F(0', s(s(z0)), s(s(z1))) -> c10(F(0', z0, f(0', s(s(z0)), s(z1))), F(0', s(s(z0)), s(z1))) The (relative) TRS S consists of the following rules: f(z0, 0', 0') -> s(z0) f(0', z0, 0') -> s(z0) f(0', 0', z0) -> s(z0) f(s(0'), z0, z1) -> f(0', s(z0), s(z1)) f(s(z0), s(z1), 0') -> f(z0, z1, s(0')) f(s(z0), 0', s(z1)) -> f(z0, s(0'), z1) f(0', s(0'), s(0')) -> s(s(0')) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0', s(s(z0)), s(0')) -> f(0', z0, s(0')) f(0', s(0'), s(s(z0))) -> f(0', s(0'), z0) f(0', s(s(z0)), s(s(z1))) -> f(0', z0, f(0', s(s(z0)), s(z1))) Rewrite Strategy: INNERMOST ---------------------------------------- (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (14) Obligation: Innermost TRS: Rules: F(z0, 0', 0') -> c F(0', z0, 0') -> c1 F(0', 0', z0) -> c2 F(s(0'), z0, z1) -> c3(F(0', s(z0), s(z1))) F(s(z0), s(z1), 0') -> c4(F(z0, z1, s(0'))) F(s(z0), 0', s(z1)) -> c5(F(z0, s(0'), z1)) F(0', s(0'), s(0')) -> c6 F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0', s(s(z0)), s(0')) -> c8(F(0', z0, s(0'))) F(0', s(0'), s(s(z0))) -> c9(F(0', s(0'), z0)) F(0', s(s(z0)), s(s(z1))) -> c10(F(0', z0, f(0', s(s(z0)), s(z1))), F(0', s(s(z0)), s(z1))) f(z0, 0', 0') -> s(z0) f(0', z0, 0') -> s(z0) f(0', 0', z0) -> s(z0) f(s(0'), z0, z1) -> f(0', s(z0), s(z1)) f(s(z0), s(z1), 0') -> f(z0, z1, s(0')) f(s(z0), 0', s(z1)) -> f(z0, s(0'), z1) f(0', s(0'), s(0')) -> s(s(0')) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0', s(s(z0)), s(0')) -> f(0', z0, s(0')) f(0', s(0'), s(s(z0))) -> f(0', s(0'), z0) f(0', s(s(z0)), s(s(z1))) -> f(0', z0, f(0', s(s(z0)), s(z1))) Types: F :: 0':s -> 0':s -> 0':s -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 0' :: 0':s c :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c1 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c2 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 s :: 0':s -> 0':s c3 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c4 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c5 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c6 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c7 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 f :: 0':s -> 0':s -> 0':s -> 0':s c8 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c9 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c10 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 hole_c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c101_11 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 hole_0':s2_11 :: 0':s gen_c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c103_11 :: Nat -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 gen_0':s4_11 :: Nat -> 0':s ---------------------------------------- (15) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: F, f They will be analysed ascendingly in the following order: f < F ---------------------------------------- (16) Obligation: Innermost TRS: Rules: F(z0, 0', 0') -> c F(0', z0, 0') -> c1 F(0', 0', z0) -> c2 F(s(0'), z0, z1) -> c3(F(0', s(z0), s(z1))) F(s(z0), s(z1), 0') -> c4(F(z0, z1, s(0'))) F(s(z0), 0', s(z1)) -> c5(F(z0, s(0'), z1)) F(0', s(0'), s(0')) -> c6 F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0', s(s(z0)), s(0')) -> c8(F(0', z0, s(0'))) F(0', s(0'), s(s(z0))) -> c9(F(0', s(0'), z0)) F(0', s(s(z0)), s(s(z1))) -> c10(F(0', z0, f(0', s(s(z0)), s(z1))), F(0', s(s(z0)), s(z1))) f(z0, 0', 0') -> s(z0) f(0', z0, 0') -> s(z0) f(0', 0', z0) -> s(z0) f(s(0'), z0, z1) -> f(0', s(z0), s(z1)) f(s(z0), s(z1), 0') -> f(z0, z1, s(0')) f(s(z0), 0', s(z1)) -> f(z0, s(0'), z1) f(0', s(0'), s(0')) -> s(s(0')) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0', s(s(z0)), s(0')) -> f(0', z0, s(0')) f(0', s(0'), s(s(z0))) -> f(0', s(0'), z0) f(0', s(s(z0)), s(s(z1))) -> f(0', z0, f(0', s(s(z0)), s(z1))) Types: F :: 0':s -> 0':s -> 0':s -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 0' :: 0':s c :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c1 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c2 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 s :: 0':s -> 0':s c3 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c4 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c5 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c6 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c7 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 f :: 0':s -> 0':s -> 0':s -> 0':s c8 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c9 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c10 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 hole_c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c101_11 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 hole_0':s2_11 :: 0':s gen_c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c103_11 :: Nat -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 gen_0':s4_11 :: Nat -> 0':s Generator Equations: gen_c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c103_11(0) <=> c gen_c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c103_11(+(x, 1)) <=> c3(gen_c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c103_11(x)) gen_0':s4_11(0) <=> 0' gen_0':s4_11(+(x, 1)) <=> s(gen_0':s4_11(x)) The following defined symbols remain to be analysed: f, F They will be analysed ascendingly in the following order: f < F ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: f(gen_0':s4_11(0), gen_0':s4_11(+(1, *(2, n6_11))), gen_0':s4_11(1)) -> gen_0':s4_11(2), rt in Omega(0) Induction Base: f(gen_0':s4_11(0), gen_0':s4_11(+(1, *(2, 0))), gen_0':s4_11(1)) ->_R^Omega(0) s(s(0')) Induction Step: f(gen_0':s4_11(0), gen_0':s4_11(+(1, *(2, +(n6_11, 1)))), gen_0':s4_11(1)) ->_R^Omega(0) f(0', gen_0':s4_11(+(1, *(2, n6_11))), s(0')) ->_IH gen_0':s4_11(2) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) Obligation: Innermost TRS: Rules: F(z0, 0', 0') -> c F(0', z0, 0') -> c1 F(0', 0', z0) -> c2 F(s(0'), z0, z1) -> c3(F(0', s(z0), s(z1))) F(s(z0), s(z1), 0') -> c4(F(z0, z1, s(0'))) F(s(z0), 0', s(z1)) -> c5(F(z0, s(0'), z1)) F(0', s(0'), s(0')) -> c6 F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0', s(s(z0)), s(0')) -> c8(F(0', z0, s(0'))) F(0', s(0'), s(s(z0))) -> c9(F(0', s(0'), z0)) F(0', s(s(z0)), s(s(z1))) -> c10(F(0', z0, f(0', s(s(z0)), s(z1))), F(0', s(s(z0)), s(z1))) f(z0, 0', 0') -> s(z0) f(0', z0, 0') -> s(z0) f(0', 0', z0) -> s(z0) f(s(0'), z0, z1) -> f(0', s(z0), s(z1)) f(s(z0), s(z1), 0') -> f(z0, z1, s(0')) f(s(z0), 0', s(z1)) -> f(z0, s(0'), z1) f(0', s(0'), s(0')) -> s(s(0')) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0', s(s(z0)), s(0')) -> f(0', z0, s(0')) f(0', s(0'), s(s(z0))) -> f(0', s(0'), z0) f(0', s(s(z0)), s(s(z1))) -> f(0', z0, f(0', s(s(z0)), s(z1))) Types: F :: 0':s -> 0':s -> 0':s -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 0' :: 0':s c :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c1 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c2 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 s :: 0':s -> 0':s c3 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c4 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c5 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c6 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c7 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 f :: 0':s -> 0':s -> 0':s -> 0':s c8 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c9 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 c10 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 hole_c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c101_11 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 hole_0':s2_11 :: 0':s gen_c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c103_11 :: Nat -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c10 gen_0':s4_11 :: Nat -> 0':s Lemmas: f(gen_0':s4_11(0), gen_0':s4_11(+(1, *(2, n6_11))), gen_0':s4_11(1)) -> gen_0':s4_11(2), rt in Omega(0) Generator Equations: gen_c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c103_11(0) <=> c gen_c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c103_11(+(x, 1)) <=> c3(gen_c:c1:c2:c3:c4:c5:c6:c7:c8:c9:c103_11(x)) gen_0':s4_11(0) <=> 0' gen_0':s4_11(+(x, 1)) <=> s(gen_0':s4_11(x)) The following defined symbols remain to be analysed: F ---------------------------------------- (19) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(z0, 0, 0) -> c F(0, z0, 0) -> c1 F(0, 0, z0) -> c2 F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(0), s(0)) -> c6 F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) S tuples: F(z0, 0, 0) -> c F(0, z0, 0) -> c1 F(0, 0, z0) -> c2 F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(0), s(0)) -> c6 F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c, c1, c2, c3_1, c4_1, c5_1, c6, c7_2, c8_1, c9_1, c10_2 ---------------------------------------- (21) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: F(0, 0, z0) -> c2 F(0, z0, 0) -> c1 F(0, s(0), s(0)) -> c6 F(z0, 0, 0) -> c ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) S tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c3_1, c4_1, c5_1, c7_2, c8_1, c9_1, c10_2 ---------------------------------------- (23) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (24) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) The (relative) TRS S consists of the following rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Rewrite Strategy: INNERMOST ---------------------------------------- (25) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (26) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (27) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (28) 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: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) [1] F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) [1] F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) [1] F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) [1] F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) [1] F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) [1] F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) [1] f(z0, 0, 0) -> s(z0) [0] f(0, z0, 0) -> s(z0) [0] f(0, 0, z0) -> s(z0) [0] f(s(0), z0, z1) -> f(0, s(z0), s(z1)) [0] f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) [0] f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) [0] f(0, s(0), s(0)) -> s(s(0)) [0] f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) [0] f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) [0] f(0, s(0), s(s(z0))) -> f(0, s(0), z0) [0] f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (29) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (30) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) [1] F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) [1] F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) [1] F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) [1] F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) [1] F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) [1] F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) [1] f(z0, 0, 0) -> s(z0) [0] f(0, z0, 0) -> s(z0) [0] f(0, 0, z0) -> s(z0) [0] f(s(0), z0, z1) -> f(0, s(z0), s(z1)) [0] f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) [0] f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) [0] f(0, s(0), s(0)) -> s(s(0)) [0] f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) [0] f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) [0] f(0, s(0), s(s(z0))) -> f(0, s(0), z0) [0] f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) [0] The TRS has the following type information: F :: 0:s -> 0:s -> 0:s -> c3:c4:c5:c7:c8:c9:c10 s :: 0:s -> 0:s 0 :: 0:s c3 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 c4 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 c5 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 c7 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 f :: 0:s -> 0:s -> 0:s -> 0:s c8 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 c9 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 c10 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 Rewrite Strategy: INNERMOST ---------------------------------------- (31) 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: f(v0, v1, v2) -> null_f [0] F(v0, v1, v2) -> null_F [0] And the following fresh constants: null_f, null_F ---------------------------------------- (32) 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: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) [1] F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) [1] F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) [1] F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) [1] F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) [1] F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) [1] F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) [1] f(z0, 0, 0) -> s(z0) [0] f(0, z0, 0) -> s(z0) [0] f(0, 0, z0) -> s(z0) [0] f(s(0), z0, z1) -> f(0, s(z0), s(z1)) [0] f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) [0] f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) [0] f(0, s(0), s(0)) -> s(s(0)) [0] f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) [0] f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) [0] f(0, s(0), s(s(z0))) -> f(0, s(0), z0) [0] f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) [0] f(v0, v1, v2) -> null_f [0] F(v0, v1, v2) -> null_F [0] The TRS has the following type information: F :: 0:s:null_f -> 0:s:null_f -> 0:s:null_f -> c3:c4:c5:c7:c8:c9:c10:null_F s :: 0:s:null_f -> 0:s:null_f 0 :: 0:s:null_f c3 :: c3:c4:c5:c7:c8:c9:c10:null_F -> c3:c4:c5:c7:c8:c9:c10:null_F c4 :: c3:c4:c5:c7:c8:c9:c10:null_F -> c3:c4:c5:c7:c8:c9:c10:null_F c5 :: c3:c4:c5:c7:c8:c9:c10:null_F -> c3:c4:c5:c7:c8:c9:c10:null_F c7 :: c3:c4:c5:c7:c8:c9:c10:null_F -> c3:c4:c5:c7:c8:c9:c10:null_F -> c3:c4:c5:c7:c8:c9:c10:null_F f :: 0:s:null_f -> 0:s:null_f -> 0:s:null_f -> 0:s:null_f c8 :: c3:c4:c5:c7:c8:c9:c10:null_F -> c3:c4:c5:c7:c8:c9:c10:null_F c9 :: c3:c4:c5:c7:c8:c9:c10:null_F -> c3:c4:c5:c7:c8:c9:c10:null_F c10 :: c3:c4:c5:c7:c8:c9:c10:null_F -> c3:c4:c5:c7:c8:c9:c10:null_F -> c3:c4:c5:c7:c8:c9:c10:null_F null_f :: 0:s:null_f null_F :: c3:c4:c5:c7:c8:c9:c10:null_F Rewrite Strategy: INNERMOST ---------------------------------------- (33) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 null_f => 0 null_F => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: F(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 F(z, z', z'') -{ 1 }-> 1 + F(z0, z1, 1 + 0) :|: z'' = 0, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 F(z, z', z'') -{ 1 }-> 1 + F(z0, 1 + 0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z'' = 1 + z1, z' = 0 F(z, z', z'') -{ 1 }-> 1 + F(0, z0, 1 + 0) :|: z0 >= 0, z = 0, z'' = 1 + 0, z' = 1 + (1 + z0) F(z, z', z'') -{ 1 }-> 1 + F(0, 1 + z0, 1 + z1) :|: z1 >= 0, z = 1 + 0, z0 >= 0, z' = z0, z'' = z1 F(z, z', z'') -{ 1 }-> 1 + F(0, 1 + 0, z0) :|: z'' = 1 + (1 + z0), z' = 1 + 0, z0 >= 0, z = 0 F(z, z', z'') -{ 1 }-> 1 + F(z0, z1, f(1 + z0, 1 + z1, z2)) + F(1 + z0, 1 + z1, z2) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z'' = 1 + z2, z2 >= 0 F(z, z', z'') -{ 1 }-> 1 + F(0, z0, f(0, 1 + (1 + z0), 1 + z1)) + F(0, 1 + (1 + z0), 1 + z1) :|: z1 >= 0, z'' = 1 + (1 + z1), z0 >= 0, z = 0, z' = 1 + (1 + z0) f(z, z', z'') -{ 0 }-> f(z0, z1, f(1 + z0, 1 + z1, z2)) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z'' = 1 + z2, z2 >= 0 f(z, z', z'') -{ 0 }-> f(z0, z1, 1 + 0) :|: z'' = 0, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 f(z, z', z'') -{ 0 }-> f(z0, 1 + 0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z'' = 1 + z1, z' = 0 f(z, z', z'') -{ 0 }-> f(0, z0, f(0, 1 + (1 + z0), 1 + z1)) :|: z1 >= 0, z'' = 1 + (1 + z1), z0 >= 0, z = 0, z' = 1 + (1 + z0) f(z, z', z'') -{ 0 }-> f(0, z0, 1 + 0) :|: z0 >= 0, z = 0, z'' = 1 + 0, z' = 1 + (1 + z0) f(z, z', z'') -{ 0 }-> f(0, 1 + z0, 1 + z1) :|: z1 >= 0, z = 1 + 0, z0 >= 0, z' = z0, z'' = z1 f(z, z', z'') -{ 0 }-> f(0, 1 + 0, z0) :|: z'' = 1 + (1 + z0), z' = 1 + 0, z0 >= 0, z = 0 f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 f(z, z', z'') -{ 0 }-> 1 + z0 :|: z'' = 0, z = z0, z0 >= 0, z' = 0 f(z, z', z'') -{ 0 }-> 1 + z0 :|: z'' = 0, z0 >= 0, z = 0, z' = z0 f(z, z', z'') -{ 0 }-> 1 + z0 :|: z'' = z0, z0 >= 0, z = 0, z' = 0 f(z, z', z'') -{ 0 }-> 1 + (1 + 0) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (35) 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: F_3 (c) The following functions are completely defined: f_3 Due to the following rules being added: f(v0, v1, v2) -> 0 [0] And the following fresh constants: const ---------------------------------------- (36) 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: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) [1] F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) [1] F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) [1] F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) [1] F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) [1] F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) [1] F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) [1] f(z0, 0, 0) -> s(z0) [0] f(0, z0, 0) -> s(z0) [0] f(0, 0, z0) -> s(z0) [0] f(s(0), z0, z1) -> f(0, s(z0), s(z1)) [0] f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) [0] f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) [0] f(0, s(0), s(0)) -> s(s(0)) [0] f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) [0] f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) [0] f(0, s(0), s(s(z0))) -> f(0, s(0), z0) [0] f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) [0] f(v0, v1, v2) -> 0 [0] The TRS has the following type information: F :: 0:s -> 0:s -> 0:s -> c3:c4:c5:c7:c8:c9:c10 s :: 0:s -> 0:s 0 :: 0:s c3 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 c4 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 c5 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 c7 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 f :: 0:s -> 0:s -> 0:s -> 0:s c8 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 c9 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 c10 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 const :: c3:c4:c5:c7:c8:c9:c10 Rewrite Strategy: INNERMOST ---------------------------------------- (37) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (38) 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: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) [1] F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) [1] F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) [1] F(s(0), s(z1), s(z2)) -> c7(F(0, z1, f(0, s(s(z1)), s(z2))), F(s(0), s(z1), z2)) [1] F(s(z0), s(z1), s(0)) -> c7(F(z0, z1, f(z0, z1, s(0))), F(s(z0), s(z1), 0)) [1] F(s(z0), s(z1), s(s(z2'))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2'))), F(s(z0), s(z1), s(z2'))) [1] F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, 0), F(s(z0), s(z1), z2)) [1] F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) [1] F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) [1] F(0, s(s(z0)), s(s(0))) -> c10(F(0, z0, f(0, z0, s(0))), F(0, s(s(z0)), s(0))) [1] F(0, s(s(z0)), s(s(s(z1')))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1')))), F(0, s(s(z0)), s(s(z1')))) [1] F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, 0), F(0, s(s(z0)), s(z1))) [1] f(z0, 0, 0) -> s(z0) [0] f(0, z0, 0) -> s(z0) [0] f(0, 0, z0) -> s(z0) [0] f(s(0), z0, z1) -> f(0, s(z0), s(z1)) [0] f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) [0] f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) [0] f(0, s(0), s(0)) -> s(s(0)) [0] f(s(0), s(z1), s(z2)) -> f(0, z1, f(0, s(s(z1)), s(z2))) [0] f(s(z0), s(z1), s(0)) -> f(z0, z1, f(z0, z1, s(0))) [0] f(s(z0), s(z1), s(s(z2''))) -> f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2''))) [0] f(s(z0), s(z1), s(z2)) -> f(z0, z1, 0) [0] f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) [0] f(0, s(0), s(s(z0))) -> f(0, s(0), z0) [0] f(0, s(s(z0)), s(s(0))) -> f(0, z0, f(0, z0, s(0))) [0] f(0, s(s(z0)), s(s(s(z1'')))) -> f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1'')))) [0] f(0, s(s(z0)), s(s(z1))) -> f(0, z0, 0) [0] f(v0, v1, v2) -> 0 [0] The TRS has the following type information: F :: 0:s -> 0:s -> 0:s -> c3:c4:c5:c7:c8:c9:c10 s :: 0:s -> 0:s 0 :: 0:s c3 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 c4 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 c5 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 c7 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 f :: 0:s -> 0:s -> 0:s -> 0:s c8 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 c9 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 c10 :: c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 -> c3:c4:c5:c7:c8:c9:c10 const :: c3:c4:c5:c7:c8:c9:c10 Rewrite Strategy: INNERMOST ---------------------------------------- (39) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 const => 0 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: F(z, z', z'') -{ 1 }-> 1 + F(z0, z1, 1 + 0) :|: z'' = 0, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 F(z, z', z'') -{ 1 }-> 1 + F(z0, 1 + 0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z'' = 1 + z1, z' = 0 F(z, z', z'') -{ 1 }-> 1 + F(0, z0, 1 + 0) :|: z0 >= 0, z = 0, z'' = 1 + 0, z' = 1 + (1 + z0) F(z, z', z'') -{ 1 }-> 1 + F(0, 1 + z0, 1 + z1) :|: z1 >= 0, z = 1 + 0, z0 >= 0, z' = z0, z'' = z1 F(z, z', z'') -{ 1 }-> 1 + F(0, 1 + 0, z0) :|: z'' = 1 + (1 + z0), z' = 1 + 0, z0 >= 0, z = 0 F(z, z', z'') -{ 1 }-> 1 + F(z0, z1, f(z0, z1, f(1 + z0, 1 + z1, z2'))) + F(1 + z0, 1 + z1, 1 + z2') :|: z1 >= 0, z'' = 1 + (1 + z2'), z = 1 + z0, z2' >= 0, z0 >= 0, z' = 1 + z1 F(z, z', z'') -{ 1 }-> 1 + F(z0, z1, f(z0, z1, 1 + 0)) + F(1 + z0, 1 + z1, 0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z'' = 1 + 0 F(z, z', z'') -{ 1 }-> 1 + F(z0, z1, 0) + F(1 + z0, 1 + z1, z2) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z'' = 1 + z2, z2 >= 0 F(z, z', z'') -{ 1 }-> 1 + F(0, z0, f(0, z0, f(0, 1 + (1 + z0), 1 + z1'))) + F(0, 1 + (1 + z0), 1 + (1 + z1')) :|: z'' = 1 + (1 + (1 + z1')), z1' >= 0, z0 >= 0, z = 0, z' = 1 + (1 + z0) F(z, z', z'') -{ 1 }-> 1 + F(0, z0, f(0, z0, 1 + 0)) + F(0, 1 + (1 + z0), 1 + 0) :|: z0 >= 0, z'' = 1 + (1 + 0), z = 0, z' = 1 + (1 + z0) F(z, z', z'') -{ 1 }-> 1 + F(0, z0, 0) + F(0, 1 + (1 + z0), 1 + z1) :|: z1 >= 0, z'' = 1 + (1 + z1), z0 >= 0, z = 0, z' = 1 + (1 + z0) F(z, z', z'') -{ 1 }-> 1 + F(0, z1, f(0, 1 + (1 + z1), 1 + z2)) + F(1 + 0, 1 + z1, z2) :|: z1 >= 0, z = 1 + 0, z' = 1 + z1, z'' = 1 + z2, z2 >= 0 f(z, z', z'') -{ 0 }-> f(z0, z1, f(z0, z1, f(1 + z0, 1 + z1, z2''))) :|: z'' = 1 + (1 + z2''), z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z2'' >= 0 f(z, z', z'') -{ 0 }-> f(z0, z1, f(z0, z1, 1 + 0)) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z'' = 1 + 0 f(z, z', z'') -{ 0 }-> f(z0, z1, 0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z'' = 1 + z2, z2 >= 0 f(z, z', z'') -{ 0 }-> f(z0, z1, 1 + 0) :|: z'' = 0, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 f(z, z', z'') -{ 0 }-> f(z0, 1 + 0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z'' = 1 + z1, z' = 0 f(z, z', z'') -{ 0 }-> f(0, z0, f(0, z0, f(0, 1 + (1 + z0), 1 + z1''))) :|: z'' = 1 + (1 + (1 + z1'')), z0 >= 0, z = 0, z' = 1 + (1 + z0), z1'' >= 0 f(z, z', z'') -{ 0 }-> f(0, z0, f(0, z0, 1 + 0)) :|: z0 >= 0, z'' = 1 + (1 + 0), z = 0, z' = 1 + (1 + z0) f(z, z', z'') -{ 0 }-> f(0, z0, 0) :|: z1 >= 0, z'' = 1 + (1 + z1), z0 >= 0, z = 0, z' = 1 + (1 + z0) f(z, z', z'') -{ 0 }-> f(0, z0, 1 + 0) :|: z0 >= 0, z = 0, z'' = 1 + 0, z' = 1 + (1 + z0) f(z, z', z'') -{ 0 }-> f(0, z1, f(0, 1 + (1 + z1), 1 + z2)) :|: z1 >= 0, z = 1 + 0, z' = 1 + z1, z'' = 1 + z2, z2 >= 0 f(z, z', z'') -{ 0 }-> f(0, 1 + z0, 1 + z1) :|: z1 >= 0, z = 1 + 0, z0 >= 0, z' = z0, z'' = z1 f(z, z', z'') -{ 0 }-> f(0, 1 + 0, z0) :|: z'' = 1 + (1 + z0), z' = 1 + 0, z0 >= 0, z = 0 f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 f(z, z', z'') -{ 0 }-> 1 + z0 :|: z'' = 0, z = z0, z0 >= 0, z' = 0 f(z, z', z'') -{ 0 }-> 1 + z0 :|: z'' = 0, z0 >= 0, z = 0, z' = z0 f(z, z', z'') -{ 0 }-> 1 + z0 :|: z'' = z0, z0 >= 0, z = 0, z' = 0 f(z, z', z'') -{ 0 }-> 1 + (1 + 0) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 ---------------------------------------- (41) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: F(z, z', z'') -{ 1 }-> 1 + F(0, z' - 2, 1 + 0) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 F(z, z', z'') -{ 1 }-> 1 + F(0, 1 + z', 1 + z'') :|: z'' >= 0, z = 1 + 0, z' >= 0 F(z, z', z'') -{ 1 }-> 1 + F(0, 1 + 0, z'' - 2) :|: z' = 1 + 0, z'' - 2 >= 0, z = 0 F(z, z', z'') -{ 1 }-> 1 + F(z - 1, z' - 1, 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 F(z, z', z'') -{ 1 }-> 1 + F(z - 1, 1 + 0, z'' - 1) :|: z'' - 1 >= 0, z - 1 >= 0, z' = 0 F(z, z', z'') -{ 1 }-> 1 + F(0, z' - 2, f(0, z' - 2, f(0, 1 + (1 + (z' - 2)), 1 + (z'' - 3)))) + F(0, 1 + (1 + (z' - 2)), 1 + (1 + (z'' - 3))) :|: z'' - 3 >= 0, z' - 2 >= 0, z = 0 F(z, z', z'') -{ 1 }-> 1 + F(0, z' - 2, f(0, z' - 2, 1 + 0)) + F(0, 1 + (1 + (z' - 2)), 1 + 0) :|: z' - 2 >= 0, z'' = 1 + (1 + 0), z = 0 F(z, z', z'') -{ 1 }-> 1 + F(0, z' - 2, 0) + F(0, 1 + (1 + (z' - 2)), 1 + (z'' - 2)) :|: z'' - 2 >= 0, z' - 2 >= 0, z = 0 F(z, z', z'') -{ 1 }-> 1 + F(0, z' - 1, f(0, 1 + (1 + (z' - 1)), 1 + (z'' - 1))) + F(1 + 0, 1 + (z' - 1), z'' - 1) :|: z' - 1 >= 0, z = 1 + 0, z'' - 1 >= 0 F(z, z', z'') -{ 1 }-> 1 + F(z - 1, z' - 1, f(z - 1, z' - 1, f(1 + (z - 1), 1 + (z' - 1), z'' - 2))) + F(1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 2)) :|: z' - 1 >= 0, z'' - 2 >= 0, z - 1 >= 0 F(z, z', z'') -{ 1 }-> 1 + F(z - 1, z' - 1, f(z - 1, z' - 1, 1 + 0)) + F(1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' = 1 + 0 F(z, z', z'') -{ 1 }-> 1 + F(z - 1, z' - 1, 0) + F(1 + (z - 1), 1 + (z' - 1), z'' - 1) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 f(z, z', z'') -{ 0 }-> f(0, z' - 2, f(0, z' - 2, f(0, 1 + (1 + (z' - 2)), 1 + (z'' - 3)))) :|: z' - 2 >= 0, z = 0, z'' - 3 >= 0 f(z, z', z'') -{ 0 }-> f(0, z' - 2, f(0, z' - 2, 1 + 0)) :|: z' - 2 >= 0, z'' = 1 + (1 + 0), z = 0 f(z, z', z'') -{ 0 }-> f(0, z' - 2, 0) :|: z'' - 2 >= 0, z' - 2 >= 0, z = 0 f(z, z', z'') -{ 0 }-> f(0, z' - 2, 1 + 0) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 f(z, z', z'') -{ 0 }-> f(0, z' - 1, f(0, 1 + (1 + (z' - 1)), 1 + (z'' - 1))) :|: z' - 1 >= 0, z = 1 + 0, z'' - 1 >= 0 f(z, z', z'') -{ 0 }-> f(0, 1 + z', 1 + z'') :|: z'' >= 0, z = 1 + 0, z' >= 0 f(z, z', z'') -{ 0 }-> f(0, 1 + 0, z'' - 2) :|: z' = 1 + 0, z'' - 2 >= 0, z = 0 f(z, z', z'') -{ 0 }-> f(z - 1, z' - 1, f(z - 1, z' - 1, f(1 + (z - 1), 1 + (z' - 1), z'' - 2))) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 2 >= 0 f(z, z', z'') -{ 0 }-> f(z - 1, z' - 1, f(z - 1, z' - 1, 1 + 0)) :|: z' - 1 >= 0, z - 1 >= 0, z'' = 1 + 0 f(z, z', z'') -{ 0 }-> f(z - 1, z' - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 f(z, z', z'') -{ 0 }-> f(z - 1, z' - 1, 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 f(z, z', z'') -{ 0 }-> f(z - 1, 1 + 0, z'' - 1) :|: z'' - 1 >= 0, z - 1 >= 0, z' = 0 f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 f(z, z', z'') -{ 0 }-> 1 + z :|: z'' = 0, z >= 0, z' = 0 f(z, z', z'') -{ 0 }-> 1 + z' :|: z'' = 0, z' >= 0, z = 0 f(z, z', z'') -{ 0 }-> 1 + z'' :|: z'' >= 0, z = 0, z' = 0 f(z, z', z'') -{ 0 }-> 1 + (1 + 0) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 ---------------------------------------- (43) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(0, s(s(z0)), s(s(z1))) -> c10(F(0, z0, f(0, s(s(z0)), s(z1))), F(0, s(s(z0)), s(z1))) by F(0, s(s(z0)), s(s(0))) -> c10(F(0, z0, f(0, z0, s(0))), F(0, s(s(z0)), s(0))) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(0))) -> c10(F(0, z0, f(0, z0, s(0))), F(0, s(s(z0)), s(0))) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) S tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(0))) -> c10(F(0, z0, f(0, z0, s(0))), F(0, s(s(z0)), s(0))) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c3_1, c4_1, c5_1, c7_2, c8_1, c9_1, c10_2 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(0, s(s(z0)), s(s(0))) -> c10(F(0, z0, f(0, z0, s(0))), F(0, s(s(z0)), s(0))) by F(0, s(s(0)), s(s(0))) -> c10(F(0, 0, s(s(0))), F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c10(F(0, s(0), s(s(0))), F(0, s(s(s(0))), s(0))) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) F(0, s(s(0)), s(s(0))) -> c10(F(0, 0, s(s(0))), F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c10(F(0, s(0), s(s(0))), F(0, s(s(s(0))), s(0))) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) S tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) F(0, s(s(0)), s(s(0))) -> c10(F(0, 0, s(s(0))), F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c10(F(0, s(0), s(s(0))), F(0, s(s(s(0))), s(0))) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c3_1, c4_1, c5_1, c7_2, c8_1, c9_1, c10_2, c10_1 ---------------------------------------- (47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) F(0, s(s(s(0))), s(s(0))) -> c10(F(0, s(0), s(s(0))), F(0, s(s(s(0))), s(0))) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) S tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) F(0, s(s(s(0))), s(s(0))) -> c10(F(0, s(0), s(s(0))), F(0, s(s(s(0))), s(0))) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c3_1, c4_1, c5_1, c7_2, c8_1, c9_1, c10_2, c10_1 ---------------------------------------- (49) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) S tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c3_1, c4_1, c5_1, c7_2, c8_1, c9_1, c10_2, c10_1, c_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(0, s(s(z0)), s(s(s(z1)))) -> c10(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(0, s(s(z0)), s(s(z1)))) by F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, 0, s(f(0, s(s(0)), s(x1)))), F(0, s(s(0)), s(s(x1)))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, 0, s(f(0, s(s(0)), s(x1)))), F(0, s(s(0)), s(s(x1)))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) S tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, 0, s(f(0, s(s(0)), s(x1)))), F(0, s(s(0)), s(s(x1)))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c3_1, c4_1, c5_1, c7_2, c8_1, c9_1, c10_2, c10_1, c_1 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) S tuples: F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c3_1, c4_1, c5_1, c7_2, c8_1, c9_1, c10_2, c10_1, c_1 ---------------------------------------- (55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), z0, z1) -> c3(F(0, s(z0), s(z1))) by F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) S tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c4_1, c5_1, c7_2, c8_1, c9_1, c10_2, c10_1, c_1, c3_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1), s(z2)) -> c7(F(z0, z1, f(s(z0), s(z1), z2)), F(s(z0), s(z1), z2)) by F(s(0), s(x1), s(z1)) -> c7(F(0, x1, f(0, s(s(x1)), s(z1))), F(s(0), s(x1), z1)) F(s(z0), s(z1), s(0)) -> c7(F(z0, z1, f(z0, z1, s(0))), F(s(z0), s(z1), 0)) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(x1), s(z1)) -> c7(F(0, x1, f(0, s(s(x1)), s(z1))), F(s(0), s(x1), z1)) F(s(z0), s(z1), s(0)) -> c7(F(z0, z1, f(z0, z1, s(0))), F(s(z0), s(z1), 0)) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) S tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(x1), s(z1)) -> c7(F(0, x1, f(0, s(s(x1)), s(z1))), F(s(0), s(x1), z1)) F(s(z0), s(z1), s(0)) -> c7(F(z0, z1, f(z0, z1, s(0))), F(s(z0), s(z1), 0)) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c4_1, c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0), s(x1), s(z1)) -> c7(F(0, x1, f(0, s(s(x1)), s(z1))), F(s(0), s(x1), z1)) by F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(z0), s(z1), s(0)) -> c7(F(z0, z1, f(z0, z1, s(0))), F(s(z0), s(z1), 0)) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) S tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(z0), s(z1), s(0)) -> c7(F(z0, z1, f(z0, z1, s(0))), F(s(z0), s(z1), 0)) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c4_1, c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1), s(0)) -> c7(F(z0, z1, f(z0, z1, s(0))), F(s(z0), s(z1), 0)) by F(s(0), s(0), s(0)) -> c7(F(0, 0, s(s(0))), F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(0), s(0), s(0)) -> c7(F(0, 0, s(s(0))), F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) S tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(0), s(0), s(0)) -> c7(F(0, 0, s(s(0))), F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c4_1, c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) S tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c4_1, c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1), s(s(z2))) -> c7(F(z0, z1, f(z0, z1, f(s(z0), s(z1), z2))), F(s(z0), s(z1), s(z2))) by F(s(0), s(0), s(s(x2))) -> c7(F(0, 0, s(f(s(0), s(0), x2))), F(s(0), s(0), s(x2))) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(0), s(x1), s(s(z1))) -> c7(F(0, x1, f(0, x1, f(0, s(s(x1)), s(z1)))), F(s(0), s(x1), s(z1))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(0), s(0), s(s(x2))) -> c7(F(0, 0, s(f(s(0), s(0), x2))), F(s(0), s(0), s(x2))) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) S tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(0), s(0), s(s(x2))) -> c7(F(0, 0, s(f(s(0), s(0), x2))), F(s(0), s(0), s(x2))) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c4_1, c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) S tuples: F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c4_1, c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1 ---------------------------------------- (69) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(z0), s(z1), 0) -> c4(F(z0, z1, s(0))) by F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) S tuples: F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(0)) -> c7(F(0, z0, f(0, z0, s(0))), F(s(0), s(z0), 0)) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c7(F(0, s(0), s(s(0))), F(s(0), s(s(0)), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c7(F(0, s(s(z0)), f(0, z0, s(0))), F(s(0), s(s(s(z0))), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1, c4_1 ---------------------------------------- (71) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) S tuples: F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c5_1, c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1, c4_1, c1_1 ---------------------------------------- (73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(z0), 0, s(z1)) -> c5(F(z0, s(0), z1)) by F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) S tuples: F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c8_1, c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1 ---------------------------------------- (75) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(s(z0)), s(0)) -> c8(F(0, z0, s(0))) by F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) S tuples: F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1 ---------------------------------------- (77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(s(s(0))), s(0))) F(0, s(s(0)), s(s(0))) -> c10(F(0, s(s(0)), s(0))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) S tuples: F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c9_1, c10_2, c10_1, c_1, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1 ---------------------------------------- (79) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(0), s(s(z0))) -> c9(F(0, s(0), z0)) by F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c10_1, c_1, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1 ---------------------------------------- (81) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: F(s(0), s(s(0)), s(0)) -> c1(F(0, s(0), s(s(0)))) F(0, s(s(s(0))), s(s(0))) -> c(F(0, s(0), s(s(0)))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c10_1, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1 ---------------------------------------- (83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(s(x0)), s(s(0))) -> c10(F(0, s(s(x0)), s(0))) by F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c10_1, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1 ---------------------------------------- (85) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: F(s(0), s(s(0)), s(0)) -> c3(F(0, s(s(s(0))), s(s(0)))) F(s(0), s(0), s(0)) -> c3(F(0, s(s(0)), s(s(0)))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c10_1, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1 ---------------------------------------- (87) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(s(0)), s(s(s(x1)))) -> c10(F(0, s(s(0)), s(s(x1)))) by F(0, s(s(0)), s(s(s(s(0))))) -> c10(F(0, s(s(0)), s(s(s(0))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(0))))) -> c10(F(0, s(s(0)), s(s(s(0))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(0))))) -> c10(F(0, s(s(0)), s(s(s(0))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(0, s(s(z0)), s(s(s(0)))) -> c10(F(0, z0, f(0, z0, f(0, z0, s(0)))), F(0, s(s(z0)), s(s(0)))) by F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, s(f(0, 0, s(0)))), F(0, s(s(0)), s(s(0)))) F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, f(0, 0, s(s(0)))), F(0, s(s(0)), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0)))), F(0, s(s(s(0))), s(s(0)))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(0))))) -> c10(F(0, s(s(0)), s(s(s(0))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, s(f(0, 0, s(0)))), F(0, s(s(0)), s(s(0)))) F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, f(0, 0, s(s(0)))), F(0, s(s(0)), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0)))), F(0, s(s(s(0))), s(s(0)))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(0))))) -> c10(F(0, s(s(0)), s(s(s(0))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, s(f(0, 0, s(0)))), F(0, s(s(0)), s(s(0)))) F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, f(0, 0, s(s(0)))), F(0, s(s(0)), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0)))), F(0, s(s(s(0))), s(s(0)))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: F(0, s(s(0)), s(s(s(s(0))))) -> c10(F(0, s(s(0)), s(s(s(0))))) F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, s(f(0, 0, s(0)))), F(0, s(s(0)), s(s(0)))) F(0, s(s(0)), s(s(s(0)))) -> c10(F(0, 0, f(0, 0, s(s(0)))), F(0, s(s(0)), s(s(0)))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0)))), F(0, s(s(s(0))), s(s(0)))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0)))), F(0, s(s(s(0))), s(s(0)))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(y0), 0) -> c3(F(0, s(s(y0)), s(0))) by F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (97) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: F(s(s(0)), 0, s(0)) -> c5(F(s(0), s(0), 0)) F(s(0), s(0), s(0)) -> c7(F(s(0), s(0), 0)) F(s(0), s(s(0)), s(0)) -> c1(F(s(0), s(s(0)), 0)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (99) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), 0, s(y0)) -> c3(F(0, s(0), s(s(y0)))) by F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (101) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(s(0)), s(0), 0) -> c4(F(s(0), 0, s(0))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (103) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(y0), s(0)) -> c3(F(0, s(s(y0)), s(s(0)))) by F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(y0), s(s(0))) -> c3(F(0, s(s(y0)), s(s(s(0))))) by F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(0), s(s(y0))) -> c3(F(0, s(s(0)), s(s(s(y0))))) by F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), s(s(0))))) by F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), 0))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), 0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), 0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(x0), s(x1)) -> c7(F(s(0), s(x0), x1)) by F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), 0))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), 0))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(x0), s(x1), s(0)) -> c7(F(s(x0), s(x1), 0)) by F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), 0))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), 0))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), f(0, s(0), 0))) by F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), s(s(0)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), s(s(0)))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), s(s(0)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (117) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: F(0, s(s(s(0))), s(s(s(0)))) -> c10(F(0, s(0), s(s(0)))) F(s(0), s(s(0)), s(s(0))) -> c3(F(0, s(s(s(0))), s(s(s(0))))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(x0), s(x1), s(s(x2))) -> c7(F(s(x0), s(x1), s(x2))) by F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z1), s(s(s(s(y1))))) -> c7(F(s(0), s(z1), s(s(s(y1))))) F(s(0), s(z1), s(s(s(y1)))) -> c7(F(s(0), s(z1), s(s(y1)))) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(z1), s(s(0))) -> c7(F(s(0), s(z1), s(0))) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(0)))) -> c7(F(s(0), s(z1), s(s(0)))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c7_1, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1 ---------------------------------------- (121) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(0), s(s(x2))) -> c7(F(s(0), s(0), s(x2))) by F(s(0), s(0), s(s(s(s(y1))))) -> c7(F(s(0), s(0), s(s(s(y1))))) F(s(0), s(0), s(s(s(y1)))) -> c7(F(s(0), s(0), s(s(y1)))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (123) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(y0)), s(0), 0) -> c4(F(s(y0), 0, s(0))) by F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (125) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(s(s(0))), 0, s(0)) -> c5(F(s(s(0)), s(0), 0)) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(s(s(y0))), 0) -> c4(F(0, s(s(y0)), s(0))) by F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(y0)), s(s(y1)), 0) -> c4(F(s(y0), s(y1), s(0))) by F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(s(0))), s(s(z1)), 0) -> c4(F(s(s(0)), s(z1), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(s(z1)), 0) -> c4(F(s(0), s(z1), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(z0), s(0)) -> c1(F(s(0), s(z0), 0)) by F(s(0), s(s(s(y0))), s(0)) -> c1(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), 0, s(s(s(y0)))) -> c5(F(0, s(0), s(s(y0)))) by F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(0)), 0, s(s(y1))) -> c5(F(s(0), s(0), s(y1))) by F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y1)))) -> c5(F(s(0), s(0), s(s(y1)))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(y0)), 0, s(s(0))) -> c5(F(s(y0), s(0), s(0))) by F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(y0)), 0, s(s(s(y2)))) -> c5(F(s(y0), s(0), s(s(y2)))) by F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(z1)))) -> c5(F(s(0), s(0), s(s(z1)))) F(s(s(s(0))), 0, s(s(s(z1)))) -> c5(F(s(s(0)), s(0), s(s(z1)))) F(s(s(z0)), 0, s(s(s(0)))) -> c5(F(s(z0), s(0), s(s(0)))) F(s(s(z0)), 0, s(s(s(s(y2))))) -> c5(F(s(z0), s(0), s(s(s(y2))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(s(y0))), 0, s(0)) -> c5(F(s(s(y0)), s(0), 0)) by F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (143) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(s(s(s(0)))), s(0), 0) -> c4(F(s(s(s(0))), 0, s(0))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c8_1, c9_1, c10_1, c7_1 ---------------------------------------- (145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(s(s(s(y0)))), s(0)) -> c8(F(0, s(s(y0)), s(0))) by F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c9_1, c10_1, c7_1, c8_1 ---------------------------------------- (147) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(0), s(s(s(s(y0))))) -> c9(F(0, s(0), s(s(y0)))) by F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c10_1, c7_1, c8_1, c9_1 ---------------------------------------- (149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(s(s(s(y0)))), s(s(0))) -> c10(F(0, s(s(s(s(y0)))), s(0))) by F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c10_1, c7_1, c8_1, c9_1 ---------------------------------------- (151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(0, s(s(0)), s(s(s(s(y0))))) -> c10(F(0, s(s(0)), s(s(s(y0))))) by F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c10_1, c7_1, c8_1, c9_1 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(z0), s(s(y1))) -> c7(F(s(0), s(z0), s(y1))) by F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(0), s(z0), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z0), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(s(s(0))))) -> c7(F(s(0), s(z0), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c10_1, c7_1, c8_1, c9_1 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(y0)), s(0), s(0)) -> c7(F(s(s(y0)), s(0), 0)) by F(s(s(s(s(y0)))), s(0), s(0)) -> c7(F(s(s(s(s(y0)))), s(0), 0)) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) F(s(s(s(s(y0)))), s(0), s(0)) -> c7(F(s(s(s(s(y0)))), s(0), 0)) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) F(s(s(s(s(y0)))), s(0), s(0)) -> c7(F(s(s(s(s(y0)))), s(0), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c10_1, c7_1, c8_1, c9_1 ---------------------------------------- (157) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(y0)), s(s(y1)), s(0)) -> c7(F(s(s(y0)), s(s(y1)), 0)) by F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(z1)), s(0)) -> c7(F(s(s(0)), s(s(z1)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(z1)), s(0)) -> c7(F(s(s(s(0))), s(s(z1)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(s(0))), s(s(s(s(s(y0))))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(s(y0))))), 0)) F(s(s(s(0))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(y0))), 0)) F(s(s(s(0))), s(s(s(0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(0))), 0)) F(s(s(s(0))), s(s(s(s(0)))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(0)))), 0)) F(s(s(s(s(0)))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(s(0)))), s(s(s(y0))), 0)) F(s(s(s(s(y0)))), s(s(s(0))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(0))), 0)) F(s(s(s(s(y0)))), s(s(s(s(y1)))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0)) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) F(s(s(s(s(y0)))), s(0), s(0)) -> c7(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(0))), s(s(s(s(s(y0))))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(s(y0))))), 0)) F(s(s(s(0))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(y0))), 0)) F(s(s(s(0))), s(s(s(0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(0))), 0)) F(s(s(s(0))), s(s(s(s(0)))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(0)))), 0)) F(s(s(s(s(0)))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(s(0)))), s(s(s(y0))), 0)) F(s(s(s(s(y0)))), s(s(s(0))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(0))), 0)) F(s(s(s(s(y0)))), s(s(s(s(y1)))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0)) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) F(s(s(s(s(y0)))), s(0), s(0)) -> c7(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(0))), s(s(s(s(s(y0))))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(s(y0))))), 0)) F(s(s(s(0))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(y0))), 0)) F(s(s(s(0))), s(s(s(0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(0))), 0)) F(s(s(s(0))), s(s(s(s(0)))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(0)))), 0)) F(s(s(s(s(0)))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(s(0)))), s(s(s(y0))), 0)) F(s(s(s(s(y0)))), s(s(s(0))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(0))), 0)) F(s(s(s(s(y0)))), s(s(s(s(y1)))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0)) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c10_1, c7_1, c8_1, c9_1 ---------------------------------------- (159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(z0), s(z1), s(s(s(y2)))) -> c7(F(s(z0), s(z1), s(s(y2)))) by F(s(0), s(z1), s(s(s(s(y1))))) -> c7(F(s(0), s(z1), s(s(s(y1))))) F(s(0), s(z1), s(s(s(z2)))) -> c7(F(s(0), s(z1), s(s(z2)))) F(s(s(0)), s(z1), s(s(s(z2)))) -> c7(F(s(s(0)), s(z1), s(s(z2)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(0)))) -> c7(F(s(0), s(z1), s(s(0)))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(z1), s(s(s(0)))) -> c7(F(s(s(0)), s(z1), s(s(0)))) F(s(s(y0)), s(0), s(s(s(0)))) -> c7(F(s(s(y0)), s(0), s(s(0)))) F(s(s(y0)), s(s(y1)), s(s(s(0)))) -> c7(F(s(s(y0)), s(s(y1)), s(s(0)))) F(s(s(0)), s(z1), s(s(s(s(y1))))) -> c7(F(s(s(0)), s(z1), s(s(s(y1))))) F(s(z0), s(z1), s(s(s(s(0))))) -> c7(F(s(z0), s(z1), s(s(s(0))))) F(s(z0), s(z1), s(s(s(s(s(y2)))))) -> c7(F(s(z0), s(z1), s(s(s(s(y2)))))) F(s(0), s(z1), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z1), s(s(s(s(s(y1))))))) F(s(0), s(z1), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z1), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(s(0)))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(s(0)))) F(s(s(0)), s(s(y0)), s(s(s(0)))) -> c7(F(s(s(0)), s(s(y0)), s(s(0)))) F(s(s(0)), s(s(0)), s(s(s(0)))) -> c7(F(s(s(0)), s(s(0)), s(s(0)))) F(s(s(0)), s(s(s(0))), s(s(s(0)))) -> c7(F(s(s(0)), s(s(s(0))), s(s(0)))) F(s(s(s(0))), s(s(y0)), s(s(s(0)))) -> c7(F(s(s(s(0))), s(s(y0)), s(s(0)))) F(s(s(s(y0))), s(s(0)), s(s(s(0)))) -> c7(F(s(s(s(y0))), s(s(0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1))), s(s(s(0)))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(s(0)))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(z1), s(s(s(s(s(s(s(y1)))))))) -> c7(F(s(0), s(z1), s(s(s(s(s(s(y1)))))))) F(s(0), s(z1), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(z1), s(s(s(s(s(0))))))) F(s(0), s(s(s(y0))), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(s(0))))))) F(s(0), s(s(0)), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(s(0)), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(0)))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(s(y0)))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(s(y0)))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(s(0)))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(s(0)))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, 0, 0) -> s(z0) f(0, z0, 0) -> s(z0) f(0, 0, z0) -> s(z0) f(s(0), z0, z1) -> f(0, s(z0), s(z1)) f(s(z0), s(z1), 0) -> f(z0, z1, s(0)) f(s(z0), 0, s(z1)) -> f(z0, s(0), z1) f(0, s(0), s(0)) -> s(s(0)) f(s(z0), s(z1), s(z2)) -> f(z0, z1, f(s(z0), s(z1), z2)) f(0, s(s(z0)), s(0)) -> f(0, z0, s(0)) f(0, s(0), s(s(z0))) -> f(0, s(0), z0) f(0, s(s(z0)), s(s(z1))) -> f(0, z0, f(0, s(s(z0)), s(z1))) Tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) F(s(s(s(s(y0)))), s(0), s(0)) -> c7(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(0))), s(s(s(s(s(y0))))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(s(y0))))), 0)) F(s(s(s(0))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(y0))), 0)) F(s(s(s(0))), s(s(s(0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(0))), 0)) F(s(s(s(0))), s(s(s(s(0)))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(0)))), 0)) F(s(s(s(s(0)))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(s(0)))), s(s(s(y0))), 0)) F(s(s(s(s(y0)))), s(s(s(0))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(0))), 0)) F(s(s(s(s(y0)))), s(s(s(s(y1)))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0)) F(s(s(0)), s(z1), s(s(s(0)))) -> c7(F(s(s(0)), s(z1), s(s(0)))) F(s(s(y0)), s(0), s(s(s(0)))) -> c7(F(s(s(y0)), s(0), s(s(0)))) F(s(s(y0)), s(s(y1)), s(s(s(0)))) -> c7(F(s(s(y0)), s(s(y1)), s(s(0)))) F(s(s(0)), s(z1), s(s(s(s(y1))))) -> c7(F(s(s(0)), s(z1), s(s(s(y1))))) F(s(z0), s(z1), s(s(s(s(0))))) -> c7(F(s(z0), s(z1), s(s(s(0))))) F(s(z0), s(z1), s(s(s(s(s(y2)))))) -> c7(F(s(z0), s(z1), s(s(s(s(y2)))))) F(s(s(0)), s(s(s(s(y0)))), s(s(s(0)))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(s(0)))) F(s(s(0)), s(s(y0)), s(s(s(0)))) -> c7(F(s(s(0)), s(s(y0)), s(s(0)))) F(s(s(0)), s(s(0)), s(s(s(0)))) -> c7(F(s(s(0)), s(s(0)), s(s(0)))) F(s(s(0)), s(s(s(0))), s(s(s(0)))) -> c7(F(s(s(0)), s(s(s(0))), s(s(0)))) F(s(s(s(0))), s(s(y0)), s(s(s(0)))) -> c7(F(s(s(s(0))), s(s(y0)), s(s(0)))) F(s(s(s(y0))), s(s(0)), s(s(s(0)))) -> c7(F(s(s(s(y0))), s(s(0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1))), s(s(s(0)))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(s(0)))) F(s(0), s(z1), s(s(s(s(s(s(s(y1)))))))) -> c7(F(s(0), s(z1), s(s(s(s(s(s(y1)))))))) F(s(0), s(z1), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(z1), s(s(s(s(s(0))))))) F(s(0), s(s(s(y0))), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(s(0))))))) F(s(0), s(s(0)), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(s(0)), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(0)))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(s(y0)))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(s(y0)))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(s(0)))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(s(0)))) S tuples: F(0, s(s(s(s(z0)))), s(s(0))) -> c10(F(0, s(s(z0)), f(0, z0, s(0))), F(0, s(s(s(s(z0)))), s(0))) F(0, s(s(z0)), s(s(s(s(z1))))) -> c10(F(0, z0, f(0, z0, f(0, z0, f(0, s(s(z0)), s(z1))))), F(0, s(s(z0)), s(s(s(z1))))) F(s(0), s(s(s(y0))), s(0)) -> c3(F(0, s(s(s(s(y0)))), s(s(0)))) F(s(0), s(y0), s(s(s(y1)))) -> c3(F(0, s(s(y0)), s(s(s(s(y1)))))) F(s(0), s(z0), s(s(z1))) -> c7(F(0, z0, f(0, z0, f(0, s(s(z0)), s(z1)))), F(s(0), s(z0), s(z1))) F(s(s(0)), s(z0), s(0)) -> c7(F(s(0), z0, f(0, s(z0), s(s(0)))), F(s(s(0)), s(z0), 0)) F(s(s(z0)), s(0), s(0)) -> c7(F(s(z0), 0, f(z0, s(0), 0)), F(s(s(z0)), s(0), 0)) F(s(s(z0)), s(s(z1)), s(0)) -> c7(F(s(z0), s(z1), f(z0, z1, f(s(z0), s(z1), 0))), F(s(s(z0)), s(s(z1)), 0)) F(s(s(0)), s(z0), s(s(x2))) -> c7(F(s(0), z0, f(0, s(z0), s(f(s(s(0)), s(z0), x2)))), F(s(s(0)), s(z0), s(x2))) F(s(z0), s(z1), s(s(0))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, s(0)))), F(s(z0), s(z1), s(0))) F(s(z0), s(z1), s(s(s(z2)))) -> c7(F(z0, z1, f(z0, z1, f(z0, z1, f(s(z0), s(z1), z2)))), F(s(z0), s(z1), s(s(z2)))) F(s(s(0)), s(s(s(s(y0)))), 0) -> c4(F(s(0), s(s(s(y0))), s(0))) F(s(s(0)), s(s(y0)), 0) -> c4(F(s(0), s(y0), s(0))) F(s(s(0)), s(s(0)), 0) -> c4(F(s(0), s(0), s(0))) F(s(s(0)), s(s(s(0))), 0) -> c4(F(s(0), s(s(0)), s(0))) F(s(s(s(0))), s(s(y0)), 0) -> c4(F(s(s(0)), s(y0), s(0))) F(s(s(s(y0))), s(s(0)), 0) -> c4(F(s(s(y0)), s(0), s(0))) F(s(s(s(y0))), s(s(s(y1))), 0) -> c4(F(s(s(y0)), s(s(y1)), s(0))) F(s(0), s(z0), s(0)) -> c1(F(0, z0, f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(0, s(s(z0)), f(0, z0, s(0)))) F(s(0), s(s(s(z0))), s(0)) -> c1(F(s(0), s(s(s(z0))), 0)) F(s(s(0)), 0, s(s(0))) -> c5(F(s(0), s(0), s(0))) F(s(s(0)), 0, s(s(s(0)))) -> c5(F(s(0), s(0), s(s(0)))) F(s(s(0)), 0, s(s(s(s(y1))))) -> c5(F(s(0), s(0), s(s(s(y1))))) F(s(s(0)), 0, s(s(s(y0)))) -> c5(F(s(0), s(0), s(s(y0)))) F(s(s(s(0))), 0, s(s(0))) -> c5(F(s(s(0)), s(0), s(0))) F(s(s(s(y0))), 0, s(s(0))) -> c5(F(s(s(y0)), s(0), s(0))) F(s(s(s(0))), 0, s(s(s(y1)))) -> c5(F(s(s(0)), s(0), s(s(y1)))) F(s(s(y0)), 0, s(s(s(0)))) -> c5(F(s(y0), s(0), s(s(0)))) F(s(s(y0)), 0, s(s(s(s(y2))))) -> c5(F(s(y0), s(0), s(s(s(y2))))) F(0, s(s(0)), s(s(s(s(s(y1)))))) -> c10(F(0, s(s(0)), s(s(s(s(y1)))))) F(0, s(s(s(s(z0)))), s(s(s(0)))) -> c10(F(0, s(s(z0)), f(0, s(s(z0)), f(0, z0, s(0)))), F(0, s(s(s(s(z0)))), s(s(0)))) F(s(0), s(s(s(y0))), 0) -> c3(F(0, s(s(s(s(y0)))), s(0))) F(s(0), 0, s(s(s(y0)))) -> c3(F(0, s(0), s(s(s(s(y0)))))) F(s(0), s(s(s(y0))), s(s(0))) -> c3(F(0, s(s(s(s(y0)))), s(s(s(0))))) F(s(0), s(0), s(s(s(y1)))) -> c3(F(0, s(s(0)), s(s(s(s(y1)))))) F(s(0), s(0), s(s(s(s(y0))))) -> c3(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(s(s(y0))), s(s(0))) -> c7(F(s(0), s(s(s(y0))), s(0))) F(s(0), s(z0), s(s(s(s(y1))))) -> c7(F(s(0), s(z0), s(s(s(y1))))) F(s(0), s(z0), s(s(s(y1)))) -> c7(F(s(0), s(z0), s(s(y1)))) F(s(0), s(z0), s(s(0))) -> c7(F(s(0), s(z0), s(0))) F(s(0), s(z0), s(s(s(0)))) -> c7(F(s(0), s(z0), s(s(0)))) F(s(0), s(0), s(s(s(y0)))) -> c7(F(s(0), s(0), s(s(y0)))) F(s(0), s(s(s(y0))), s(0)) -> c7(F(s(0), s(s(s(y0))), 0)) F(s(0), s(s(s(y0))), s(s(s(0)))) -> c7(F(s(0), s(s(s(y0))), s(s(0)))) F(s(0), s(s(0)), s(s(s(0)))) -> c7(F(s(0), s(s(0)), s(s(0)))) F(s(0), s(0), s(s(s(s(y0))))) -> c7(F(s(0), s(0), s(s(s(y0))))) F(s(0), s(0), s(s(s(s(s(y0)))))) -> c7(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), s(s(s(s(y0)))), s(0)) -> c7(F(s(s(0)), s(s(s(s(y0)))), 0)) F(s(s(0)), s(s(y0)), s(0)) -> c7(F(s(s(0)), s(s(y0)), 0)) F(s(s(0)), s(s(0)), s(0)) -> c7(F(s(s(0)), s(s(0)), 0)) F(s(s(0)), s(s(s(0))), s(0)) -> c7(F(s(s(0)), s(s(s(0))), 0)) F(s(s(s(0))), s(s(y0)), s(0)) -> c7(F(s(s(s(0))), s(s(y0)), 0)) F(s(s(s(y0))), s(s(0)), s(0)) -> c7(F(s(s(s(y0))), s(s(0)), 0)) F(s(s(s(y0))), s(s(s(y1))), s(0)) -> c7(F(s(s(s(y0))), s(s(s(y1))), 0)) F(s(s(0)), s(z1), s(s(0))) -> c7(F(s(s(0)), s(z1), s(0))) F(s(s(y0)), s(0), s(s(0))) -> c7(F(s(s(y0)), s(0), s(0))) F(s(s(y0)), s(s(y1)), s(s(0))) -> c7(F(s(s(y0)), s(s(y1)), s(0))) F(s(s(0)), s(z1), s(s(s(y1)))) -> c7(F(s(s(0)), s(z1), s(s(y1)))) F(s(z0), s(z1), s(s(s(0)))) -> c7(F(s(z0), s(z1), s(s(0)))) F(s(z0), s(z1), s(s(s(s(y2))))) -> c7(F(s(z0), s(z1), s(s(s(y2))))) F(s(0), s(z1), s(s(s(s(s(y1)))))) -> c7(F(s(0), s(z1), s(s(s(s(y1)))))) F(s(0), s(z1), s(s(s(s(0))))) -> c7(F(s(0), s(z1), s(s(s(0))))) F(s(0), s(s(s(y0))), s(s(s(s(0))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(0))))) F(s(0), s(s(0)), s(s(s(s(0))))) -> c7(F(s(0), s(s(0)), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(s(y0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), s(s(s(s(y0)))), s(s(0))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(0)), s(s(y0)), s(s(0))) -> c7(F(s(s(0)), s(s(y0)), s(0))) F(s(s(0)), s(s(0)), s(s(0))) -> c7(F(s(s(0)), s(s(0)), s(0))) F(s(s(0)), s(s(s(0))), s(s(0))) -> c7(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(0))), s(s(y0)), s(s(0))) -> c7(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(y0))), s(s(0)), s(s(0))) -> c7(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(y0))), s(s(s(y1))), s(s(0))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(0), s(s(s(0)))) -> c7(F(s(0), s(0), s(s(0)))) F(s(0), s(0), s(s(0))) -> c7(F(s(0), s(0), s(0))) F(s(0), s(0), s(s(s(s(0))))) -> c7(F(s(0), s(0), s(s(s(0))))) F(s(0), s(0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(0), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(s(y0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(s(s(y0)))), s(0), 0) -> c4(F(s(s(s(y0))), 0, s(0))) F(s(0), s(s(s(s(s(y0))))), 0) -> c4(F(0, s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(s(s(y0))))), 0) -> c4(F(s(s(0)), s(s(s(s(y0)))), s(0))) F(s(s(s(0))), s(s(s(y0))), 0) -> c4(F(s(s(0)), s(s(y0)), s(0))) F(s(s(s(0))), s(s(s(0))), 0) -> c4(F(s(s(0)), s(s(0)), s(0))) F(s(s(s(0))), s(s(s(s(0)))), 0) -> c4(F(s(s(0)), s(s(s(0))), s(0))) F(s(s(s(s(0)))), s(s(s(y0))), 0) -> c4(F(s(s(s(0))), s(s(y0)), s(0))) F(s(s(s(s(y0)))), s(s(s(0))), 0) -> c4(F(s(s(s(y0))), s(s(0)), s(0))) F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0) -> c4(F(s(s(s(y0))), s(s(s(y1))), s(0))) F(s(0), s(s(s(s(s(y0))))), s(0)) -> c1(F(s(0), s(s(s(s(s(y0))))), 0)) F(s(0), 0, s(s(s(s(s(y0)))))) -> c5(F(0, s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(s(y0)))))) -> c5(F(s(0), s(0), s(s(s(s(y0)))))) F(s(s(0)), 0, s(s(s(s(0))))) -> c5(F(s(0), s(0), s(s(s(0))))) F(s(s(0)), 0, s(s(s(s(s(s(y0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(y0))))))) F(s(s(0)), 0, s(s(s(s(s(0)))))) -> c5(F(s(0), s(0), s(s(s(s(0)))))) F(s(s(0)), 0, s(s(s(s(s(s(s(y0)))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(y0)))))))) F(s(s(0)), 0, s(s(s(s(s(s(0))))))) -> c5(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(s(0)), 0, s(s(s(s(s(s(s(s(y0))))))))) -> c5(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(s(s(0))), 0, s(s(s(0)))) -> c5(F(s(s(0)), s(0), s(s(0)))) F(s(s(s(y0))), 0, s(s(s(0)))) -> c5(F(s(s(y0)), s(0), s(s(0)))) F(s(s(s(0))), 0, s(s(s(s(y1))))) -> c5(F(s(s(0)), s(0), s(s(s(y1))))) F(s(s(z0)), 0, s(s(s(s(0))))) -> c5(F(s(z0), s(0), s(s(s(0))))) F(s(s(z0)), 0, s(s(s(s(s(y2)))))) -> c5(F(s(z0), s(0), s(s(s(s(y2)))))) F(s(s(s(s(s(y0))))), 0, s(0)) -> c5(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(s(s(0))))), 0, s(0)) -> c5(F(s(s(s(s(0)))), s(0), 0)) F(0, s(s(s(s(s(s(y0)))))), s(0)) -> c8(F(0, s(s(s(s(y0)))), s(0))) F(0, s(0), s(s(s(s(s(s(y0))))))) -> c9(F(0, s(0), s(s(s(s(y0)))))) F(0, s(s(s(s(s(s(y0)))))), s(s(0))) -> c10(F(0, s(s(s(s(s(s(y0)))))), s(0))) F(0, s(s(0)), s(s(s(s(s(s(y0))))))) -> c10(F(0, s(s(0)), s(s(s(s(s(y0))))))) F(s(0), s(z0), s(s(s(s(s(s(y1))))))) -> c7(F(s(0), s(z0), s(s(s(s(s(y1))))))) F(s(0), s(z0), s(s(s(s(s(0)))))) -> c7(F(s(0), s(z0), s(s(s(s(0)))))) F(s(0), s(s(s(y0))), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(0)))))) F(s(0), s(s(0)), s(s(s(s(s(0)))))) -> c7(F(s(0), s(s(0)), s(s(s(s(0)))))) F(s(0), s(0), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(0), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(y0))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(y0))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(0))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(0))) F(s(s(s(s(y0)))), s(0), s(0)) -> c7(F(s(s(s(s(y0)))), s(0), 0)) F(s(s(s(0))), s(s(s(s(s(y0))))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(s(y0))))), 0)) F(s(s(s(0))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(y0))), 0)) F(s(s(s(0))), s(s(s(0))), s(0)) -> c7(F(s(s(s(0))), s(s(s(0))), 0)) F(s(s(s(0))), s(s(s(s(0)))), s(0)) -> c7(F(s(s(s(0))), s(s(s(s(0)))), 0)) F(s(s(s(s(0)))), s(s(s(y0))), s(0)) -> c7(F(s(s(s(s(0)))), s(s(s(y0))), 0)) F(s(s(s(s(y0)))), s(s(s(0))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(0))), 0)) F(s(s(s(s(y0)))), s(s(s(s(y1)))), s(0)) -> c7(F(s(s(s(s(y0)))), s(s(s(s(y1)))), 0)) F(s(s(0)), s(z1), s(s(s(0)))) -> c7(F(s(s(0)), s(z1), s(s(0)))) F(s(s(y0)), s(0), s(s(s(0)))) -> c7(F(s(s(y0)), s(0), s(s(0)))) F(s(s(y0)), s(s(y1)), s(s(s(0)))) -> c7(F(s(s(y0)), s(s(y1)), s(s(0)))) F(s(s(0)), s(z1), s(s(s(s(y1))))) -> c7(F(s(s(0)), s(z1), s(s(s(y1))))) F(s(z0), s(z1), s(s(s(s(0))))) -> c7(F(s(z0), s(z1), s(s(s(0))))) F(s(z0), s(z1), s(s(s(s(s(y2)))))) -> c7(F(s(z0), s(z1), s(s(s(s(y2)))))) F(s(s(0)), s(s(s(s(y0)))), s(s(s(0)))) -> c7(F(s(s(0)), s(s(s(s(y0)))), s(s(0)))) F(s(s(0)), s(s(y0)), s(s(s(0)))) -> c7(F(s(s(0)), s(s(y0)), s(s(0)))) F(s(s(0)), s(s(0)), s(s(s(0)))) -> c7(F(s(s(0)), s(s(0)), s(s(0)))) F(s(s(0)), s(s(s(0))), s(s(s(0)))) -> c7(F(s(s(0)), s(s(s(0))), s(s(0)))) F(s(s(s(0))), s(s(y0)), s(s(s(0)))) -> c7(F(s(s(s(0))), s(s(y0)), s(s(0)))) F(s(s(s(y0))), s(s(0)), s(s(s(0)))) -> c7(F(s(s(s(y0))), s(s(0)), s(s(0)))) F(s(s(s(y0))), s(s(s(y1))), s(s(s(0)))) -> c7(F(s(s(s(y0))), s(s(s(y1))), s(s(0)))) F(s(0), s(z1), s(s(s(s(s(s(s(y1)))))))) -> c7(F(s(0), s(z1), s(s(s(s(s(s(y1)))))))) F(s(0), s(z1), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(z1), s(s(s(s(s(0))))))) F(s(0), s(s(s(y0))), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(s(s(y0))), s(s(s(s(s(0))))))) F(s(0), s(s(0)), s(s(s(s(s(s(0))))))) -> c7(F(s(0), s(s(0)), s(s(s(s(s(0))))))) F(s(0), s(0), s(s(s(s(s(s(s(0)))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(0)))))))) F(s(0), s(0), s(s(s(s(s(s(s(s(s(y0)))))))))) -> c7(F(s(0), s(0), s(s(s(s(s(s(s(s(y0)))))))))) F(s(0), s(s(s(s(s(y0))))), s(s(s(0)))) -> c7(F(s(0), s(s(s(s(s(y0))))), s(s(0)))) K tuples:none Defined Rule Symbols: f_3 Defined Pair Symbols: F_3 Compound Symbols: c10_2, c3_1, c7_2, c4_1, c1_1, c5_1, c10_1, c7_1, c8_1, c9_1 ---------------------------------------- (161) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (162) 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: f(x, 0, 0) -> s(x) [1] f(0, y, 0) -> s(y) [1] f(0, 0, z) -> s(z) [1] f(s(0), y, z) -> f(0, s(y), s(z)) [1] f(s(x), s(y), 0) -> f(x, y, s(0)) [1] f(s(x), 0, s(z)) -> f(x, s(0), z) [1] f(0, s(0), s(0)) -> s(s(0)) [1] f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) [1] f(0, s(s(y)), s(0)) -> f(0, y, s(0)) [1] f(0, s(0), s(s(z))) -> f(0, s(0), z) [1] f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (163) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (164) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(x, 0, 0) -> s(x) [1] f(0, y, 0) -> s(y) [1] f(0, 0, z) -> s(z) [1] f(s(0), y, z) -> f(0, s(y), s(z)) [1] f(s(x), s(y), 0) -> f(x, y, s(0)) [1] f(s(x), 0, s(z)) -> f(x, s(0), z) [1] f(0, s(0), s(0)) -> s(s(0)) [1] f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) [1] f(0, s(s(y)), s(0)) -> f(0, y, s(0)) [1] f(0, s(0), s(s(z))) -> f(0, s(0), z) [1] f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) [1] The TRS has the following type information: f :: 0:s -> 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (165) 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: none (c) The following functions are completely defined: f_3 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (166) 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: f(x, 0, 0) -> s(x) [1] f(0, y, 0) -> s(y) [1] f(0, 0, z) -> s(z) [1] f(s(0), y, z) -> f(0, s(y), s(z)) [1] f(s(x), s(y), 0) -> f(x, y, s(0)) [1] f(s(x), 0, s(z)) -> f(x, s(0), z) [1] f(0, s(0), s(0)) -> s(s(0)) [1] f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) [1] f(0, s(s(y)), s(0)) -> f(0, y, s(0)) [1] f(0, s(0), s(s(z))) -> f(0, s(0), z) [1] f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) [1] The TRS has the following type information: f :: 0:s -> 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (167) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (168) 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: f(x, 0, 0) -> s(x) [1] f(0, y, 0) -> s(y) [1] f(0, 0, z) -> s(z) [1] f(s(0), y, z) -> f(0, s(y), s(z)) [1] f(s(x), s(y), 0) -> f(x, y, s(0)) [1] f(s(x), 0, s(z)) -> f(x, s(0), z) [1] f(0, s(0), s(0)) -> s(s(0)) [1] f(s(0), s(y), s(z)) -> f(0, y, f(0, s(s(y)), s(z))) [2] f(s(x), s(y), s(0)) -> f(x, y, f(x, y, s(0))) [2] f(s(x), s(y), s(s(z'))) -> f(x, y, f(x, y, f(s(x), s(y), z'))) [2] f(0, s(s(y)), s(0)) -> f(0, y, s(0)) [1] f(0, s(0), s(s(z))) -> f(0, s(0), z) [1] f(0, s(s(y)), s(s(0))) -> f(0, y, f(0, y, s(0))) [2] f(0, s(s(y)), s(s(s(z'')))) -> f(0, y, f(0, y, f(0, s(s(y)), s(z'')))) [2] The TRS has the following type information: f :: 0:s -> 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (169) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 ---------------------------------------- (170) Obligation: Complexity RNTS consisting of the following rules: f(z1, z2, z3) -{ 2 }-> f(x, y, f(x, y, f(1 + x, 1 + y, z'))) :|: z2 = 1 + y, x >= 0, y >= 0, z' >= 0, z1 = 1 + x, z3 = 1 + (1 + z') f(z1, z2, z3) -{ 2 }-> f(x, y, f(x, y, 1 + 0)) :|: z2 = 1 + y, x >= 0, y >= 0, z1 = 1 + x, z3 = 1 + 0 f(z1, z2, z3) -{ 1 }-> f(x, y, 1 + 0) :|: z2 = 1 + y, z3 = 0, x >= 0, y >= 0, z1 = 1 + x f(z1, z2, z3) -{ 1 }-> f(x, 1 + 0, z) :|: z >= 0, z2 = 0, x >= 0, z3 = 1 + z, z1 = 1 + x f(z1, z2, z3) -{ 2 }-> f(0, y, f(0, y, f(0, 1 + (1 + y), 1 + z''))) :|: z2 = 1 + (1 + y), z1 = 0, z'' >= 0, y >= 0, z3 = 1 + (1 + (1 + z'')) f(z1, z2, z3) -{ 2 }-> f(0, y, f(0, y, 1 + 0)) :|: z2 = 1 + (1 + y), z1 = 0, z3 = 1 + (1 + 0), y >= 0 f(z1, z2, z3) -{ 2 }-> f(0, y, f(0, 1 + (1 + y), 1 + z)) :|: z >= 0, z2 = 1 + y, z1 = 1 + 0, y >= 0, z3 = 1 + z f(z1, z2, z3) -{ 1 }-> f(0, y, 1 + 0) :|: z2 = 1 + (1 + y), z1 = 0, y >= 0, z3 = 1 + 0 f(z1, z2, z3) -{ 1 }-> f(0, 1 + y, 1 + z) :|: z2 = y, z >= 0, z1 = 1 + 0, z3 = z, y >= 0 f(z1, z2, z3) -{ 1 }-> f(0, 1 + 0, z) :|: z1 = 0, z2 = 1 + 0, z >= 0, z3 = 1 + (1 + z) f(z1, z2, z3) -{ 1 }-> 1 + x :|: z2 = 0, z3 = 0, x >= 0, z1 = x f(z1, z2, z3) -{ 1 }-> 1 + y :|: z1 = 0, z2 = y, z3 = 0, y >= 0 f(z1, z2, z3) -{ 1 }-> 1 + z :|: z1 = 0, z >= 0, z2 = 0, z3 = z f(z1, z2, z3) -{ 1 }-> 1 + (1 + 0) :|: z1 = 0, z2 = 1 + 0, z3 = 1 + 0 ---------------------------------------- (171) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (172) Obligation: Complexity RNTS consisting of the following rules: f(z1, z2, z3) -{ 2 }-> f(0, z2 - 2, f(0, z2 - 2, f(0, 1 + (1 + (z2 - 2)), 1 + (z3 - 3)))) :|: z1 = 0, z3 - 3 >= 0, z2 - 2 >= 0 f(z1, z2, z3) -{ 2 }-> f(0, z2 - 2, f(0, z2 - 2, 1 + 0)) :|: z1 = 0, z3 = 1 + (1 + 0), z2 - 2 >= 0 f(z1, z2, z3) -{ 1 }-> f(0, z2 - 2, 1 + 0) :|: z1 = 0, z2 - 2 >= 0, z3 = 1 + 0 f(z1, z2, z3) -{ 2 }-> f(0, z2 - 1, f(0, 1 + (1 + (z2 - 1)), 1 + (z3 - 1))) :|: z3 - 1 >= 0, z1 = 1 + 0, z2 - 1 >= 0 f(z1, z2, z3) -{ 1 }-> f(0, 1 + z2, 1 + z3) :|: z3 >= 0, z1 = 1 + 0, z2 >= 0 f(z1, z2, z3) -{ 1 }-> f(0, 1 + 0, z3 - 2) :|: z1 = 0, z2 = 1 + 0, z3 - 2 >= 0 f(z1, z2, z3) -{ 2 }-> f(z1 - 1, z2 - 1, f(z1 - 1, z2 - 1, f(1 + (z1 - 1), 1 + (z2 - 1), z3 - 2))) :|: z1 - 1 >= 0, z2 - 1 >= 0, z3 - 2 >= 0 f(z1, z2, z3) -{ 2 }-> f(z1 - 1, z2 - 1, f(z1 - 1, z2 - 1, 1 + 0)) :|: z1 - 1 >= 0, z2 - 1 >= 0, z3 = 1 + 0 f(z1, z2, z3) -{ 1 }-> f(z1 - 1, z2 - 1, 1 + 0) :|: z3 = 0, z1 - 1 >= 0, z2 - 1 >= 0 f(z1, z2, z3) -{ 1 }-> f(z1 - 1, 1 + 0, z3 - 1) :|: z3 - 1 >= 0, z2 = 0, z1 - 1 >= 0 f(z1, z2, z3) -{ 1 }-> 1 + z1 :|: z2 = 0, z3 = 0, z1 >= 0 f(z1, z2, z3) -{ 1 }-> 1 + z2 :|: z1 = 0, z3 = 0, z2 >= 0 f(z1, z2, z3) -{ 1 }-> 1 + z3 :|: z1 = 0, z3 >= 0, z2 = 0 f(z1, z2, z3) -{ 1 }-> 1 + (1 + 0) :|: z1 = 0, z2 = 1 + 0, z3 = 1 + 0 ---------------------------------------- (173) 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: none And the following fresh constants: none ---------------------------------------- (174) 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: f(x, 0, 0) -> s(x) [1] f(0, y, 0) -> s(y) [1] f(0, 0, z) -> s(z) [1] f(s(0), y, z) -> f(0, s(y), s(z)) [1] f(s(x), s(y), 0) -> f(x, y, s(0)) [1] f(s(x), 0, s(z)) -> f(x, s(0), z) [1] f(0, s(0), s(0)) -> s(s(0)) [1] f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) [1] f(0, s(s(y)), s(0)) -> f(0, y, s(0)) [1] f(0, s(0), s(s(z))) -> f(0, s(0), z) [1] f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z))) [1] The TRS has the following type information: f :: 0:s -> 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (175) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 ---------------------------------------- (176) Obligation: Complexity RNTS consisting of the following rules: f(z', z'', z1) -{ 1 }-> f(x, y, f(1 + x, 1 + y, z)) :|: z' = 1 + x, z >= 0, x >= 0, y >= 0, z'' = 1 + y, z1 = 1 + z f(z', z'', z1) -{ 1 }-> f(x, y, 1 + 0) :|: z' = 1 + x, z1 = 0, x >= 0, y >= 0, z'' = 1 + y f(z', z'', z1) -{ 1 }-> f(x, 1 + 0, z) :|: z'' = 0, z' = 1 + x, z >= 0, x >= 0, z1 = 1 + z f(z', z'', z1) -{ 1 }-> f(0, y, f(0, 1 + (1 + y), 1 + z)) :|: z >= 0, y >= 0, z'' = 1 + (1 + y), z1 = 1 + (1 + z), z' = 0 f(z', z'', z1) -{ 1 }-> f(0, y, 1 + 0) :|: z1 = 1 + 0, y >= 0, z'' = 1 + (1 + y), z' = 0 f(z', z'', z1) -{ 1 }-> f(0, 1 + y, 1 + z) :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 1 + 0 f(z', z'', z1) -{ 1 }-> f(0, 1 + 0, z) :|: z >= 0, z1 = 1 + (1 + z), z'' = 1 + 0, z' = 0 f(z', z'', z1) -{ 1 }-> 1 + x :|: z'' = 0, z1 = 0, z' = x, x >= 0 f(z', z'', z1) -{ 1 }-> 1 + y :|: z1 = 0, z'' = y, y >= 0, z' = 0 f(z', z'', z1) -{ 1 }-> 1 + z :|: z'' = 0, z1 = z, z >= 0, z' = 0 f(z', z'', z1) -{ 1 }-> 1 + (1 + 0) :|: z1 = 1 + 0, z'' = 1 + 0, z' = 0 Only complete derivations are relevant for the runtime complexity.