KILLED proof of input_ztVGFpZTDE.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxRelTRS (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (12) typed CpxTrs (13) OrderProof [LOWER BOUND(ID), 6 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 295 ms] (16) BEST (17) proven lower bound (18) LowerBoundPropagationProof [FINISHED, 0 ms] (19) BOUNDS(n^1, INF) (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 126 ms] (22) typed CpxTrs (23) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (24) CdtProblem (25) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CpxRelTRS (33) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (34) CpxTRS (35) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CpxWeightedTrs (37) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CpxTypedWeightedTrs (39) CompletionProof [UPPER BOUND(ID), 0 ms] (40) CpxTypedWeightedCompleteTrs (41) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CpxTypedWeightedCompleteTrs (43) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) InliningProof [UPPER BOUND(ID), 154 ms] (46) CpxRNTS (47) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CpxRNTS (49) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CpxRNTS (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 154 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 53 ms] (56) CpxRNTS (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 122 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 90 ms] (66) CpxRNTS (67) IntTrsBoundProof [UPPER BOUND(ID), 4 ms] (68) CpxRNTS (69) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 2855 ms] (72) CpxRNTS (73) IntTrsBoundProof [UPPER BOUND(ID), 70 ms] (74) CpxRNTS (75) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 345 ms] (78) CpxRNTS (79) IntTrsBoundProof [UPPER BOUND(ID), 167 ms] (80) CpxRNTS (81) CompletionProof [UPPER BOUND(ID), 0 ms] (82) CpxTypedWeightedCompleteTrs (83) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (84) CpxRNTS (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 122 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 35 ms] (108) CdtProblem (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 68 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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 19 ms] (144) CdtProblem (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 5 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), 31 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 9 ms] (156) CdtProblem (157) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 18 ms] (160) CdtProblem (161) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 19 ms] (174) CdtProblem (175) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 36 ms] (180) CdtProblem (181) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 91 ms] (184) CdtProblem (185) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 185 ms] (186) CdtProblem (187) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 68 ms] (188) CdtProblem (189) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 13 ms] (190) CdtProblem (191) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 27 ms] (192) CdtProblem (193) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (194) CpxWeightedTrs (195) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CpxTypedWeightedTrs (197) CompletionProof [UPPER BOUND(ID), 0 ms] (198) CpxTypedWeightedCompleteTrs (199) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CpxTypedWeightedCompleteTrs (201) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (202) CpxRNTS (203) InliningProof [UPPER BOUND(ID), 88 ms] (204) CpxRNTS (205) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CpxRNTS (207) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CpxRNTS (209) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (210) CpxRNTS (211) IntTrsBoundProof [UPPER BOUND(ID), 238 ms] (212) CpxRNTS (213) IntTrsBoundProof [UPPER BOUND(ID), 9 ms] (214) CpxRNTS (215) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (216) CpxRNTS (217) IntTrsBoundProof [UPPER BOUND(ID), 172 ms] (218) CpxRNTS (219) IntTrsBoundProof [UPPER BOUND(ID), 53 ms] (220) CpxRNTS (221) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (222) CpxRNTS (223) IntTrsBoundProof [UPPER BOUND(ID), 2323 ms] (224) CpxRNTS (225) IntTrsBoundProof [UPPER BOUND(ID), 99 ms] (226) CpxRNTS (227) CompletionProof [UPPER BOUND(ID), 0 ms] (228) CpxTypedWeightedCompleteTrs (229) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (230) 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) -> cons(X, n__f(n__g(X))) g(0) -> s(0) g(s(X)) -> s(s(g(X))) sel(0, cons(X, Y)) -> X sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) f(X) -> n__f(X) g(X) -> n__g(X) activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) activate(X) -> X 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) -> cons(X, n__f(n__g(X))) g(0') -> s(0') g(s(X)) -> s(s(g(X))) sel(0', cons(X, Y)) -> X sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) f(X) -> n__f(X) g(X) -> n__g(X) activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) activate(X) -> X 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) -> cons(X, n__f(n__g(X))) g(0) -> s(0) g(s(X)) -> s(s(g(X))) sel(0, cons(X, Y)) -> X sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) f(X) -> n__f(X) g(X) -> n__g(X) activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) activate(X) -> X 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) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 Tuples: F(z0) -> c F(z0) -> c1 G(0) -> c2 G(s(z0)) -> c3(G(z0)) G(z0) -> c4 SEL(0, cons(z0, z1)) -> c5 SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__f(z0)) -> c7(F(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(z0) -> c9 S tuples: F(z0) -> c F(z0) -> c1 G(0) -> c2 G(s(z0)) -> c3(G(z0)) G(z0) -> c4 SEL(0, cons(z0, z1)) -> c5 SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__f(z0)) -> c7(F(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(z0) -> c9 K tuples:none Defined Rule Symbols: f_1, g_1, sel_2, activate_1 Defined Pair Symbols: F_1, G_1, SEL_2, ACTIVATE_1 Compound Symbols: c, c1, c2, c3_1, c4, c5, c6_2, c7_2, c8_2, c9 ---------------------------------------- (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(n^1, INF). The TRS R consists of the following rules: F(z0) -> c F(z0) -> c1 G(0) -> c2 G(s(z0)) -> c3(G(z0)) G(z0) -> c4 SEL(0, cons(z0, z1)) -> c5 SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__f(z0)) -> c7(F(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(z0) -> c9 The (relative) TRS S consists of the following rules: f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 Rewrite Strategy: INNERMOST ---------------------------------------- (9) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (10) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: F(z0) -> c F(z0) -> c1 G(0') -> c2 G(s(z0)) -> c3(G(z0)) G(z0) -> c4 SEL(0', cons(z0, z1)) -> c5 SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__f(z0)) -> c7(F(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(z0) -> c9 The (relative) TRS S consists of the following rules: f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0') -> s(0') g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) sel(0', cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (12) Obligation: Innermost TRS: Rules: F(z0) -> c F(z0) -> c1 G(0') -> c2 G(s(z0)) -> c3(G(z0)) G(z0) -> c4 SEL(0', cons(z0, z1)) -> c5 SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__f(z0)) -> c7(F(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(z0) -> c9 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0') -> s(0') g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) sel(0', cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 Types: F :: 0':s:cons:n__f:n__g -> c:c1 c :: c:c1 c1 :: c:c1 G :: 0':s:cons:n__f:n__g -> c2:c3:c4 0' :: 0':s:cons:n__f:n__g c2 :: c2:c3:c4 s :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c3 :: c2:c3:c4 -> c2:c3:c4 c4 :: c2:c3:c4 SEL :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g -> c5:c6 cons :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c5 :: c5:c6 c6 :: c5:c6 -> c7:c8:c9 -> c5:c6 activate :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g ACTIVATE :: 0':s:cons:n__f:n__g -> c7:c8:c9 n__f :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c7 :: c:c1 -> c7:c8:c9 -> c7:c8:c9 n__g :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c8 :: c2:c3:c4 -> c7:c8:c9 -> c7:c8:c9 c9 :: c7:c8:c9 f :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g g :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g sel :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g hole_c:c11_10 :: c:c1 hole_0':s:cons:n__f:n__g2_10 :: 0':s:cons:n__f:n__g hole_c2:c3:c43_10 :: c2:c3:c4 hole_c5:c64_10 :: c5:c6 hole_c7:c8:c95_10 :: c7:c8:c9 gen_0':s:cons:n__f:n__g6_10 :: Nat -> 0':s:cons:n__f:n__g gen_c2:c3:c47_10 :: Nat -> c2:c3:c4 gen_c5:c68_10 :: Nat -> c5:c6 gen_c7:c8:c99_10 :: Nat -> c7:c8:c9 ---------------------------------------- (13) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: G, SEL, activate, ACTIVATE, g, sel They will be analysed ascendingly in the following order: G < ACTIVATE activate < SEL ACTIVATE < SEL activate < ACTIVATE g < activate activate < sel ---------------------------------------- (14) Obligation: Innermost TRS: Rules: F(z0) -> c F(z0) -> c1 G(0') -> c2 G(s(z0)) -> c3(G(z0)) G(z0) -> c4 SEL(0', cons(z0, z1)) -> c5 SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__f(z0)) -> c7(F(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(z0) -> c9 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0') -> s(0') g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) sel(0', cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 Types: F :: 0':s:cons:n__f:n__g -> c:c1 c :: c:c1 c1 :: c:c1 G :: 0':s:cons:n__f:n__g -> c2:c3:c4 0' :: 0':s:cons:n__f:n__g c2 :: c2:c3:c4 s :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c3 :: c2:c3:c4 -> c2:c3:c4 c4 :: c2:c3:c4 SEL :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g -> c5:c6 cons :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c5 :: c5:c6 c6 :: c5:c6 -> c7:c8:c9 -> c5:c6 activate :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g ACTIVATE :: 0':s:cons:n__f:n__g -> c7:c8:c9 n__f :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c7 :: c:c1 -> c7:c8:c9 -> c7:c8:c9 n__g :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c8 :: c2:c3:c4 -> c7:c8:c9 -> c7:c8:c9 c9 :: c7:c8:c9 f :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g g :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g sel :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g hole_c:c11_10 :: c:c1 hole_0':s:cons:n__f:n__g2_10 :: 0':s:cons:n__f:n__g hole_c2:c3:c43_10 :: c2:c3:c4 hole_c5:c64_10 :: c5:c6 hole_c7:c8:c95_10 :: c7:c8:c9 gen_0':s:cons:n__f:n__g6_10 :: Nat -> 0':s:cons:n__f:n__g gen_c2:c3:c47_10 :: Nat -> c2:c3:c4 gen_c5:c68_10 :: Nat -> c5:c6 gen_c7:c8:c99_10 :: Nat -> c7:c8:c9 Generator Equations: gen_0':s:cons:n__f:n__g6_10(0) <=> 0' gen_0':s:cons:n__f:n__g6_10(+(x, 1)) <=> s(gen_0':s:cons:n__f:n__g6_10(x)) gen_c2:c3:c47_10(0) <=> c2 gen_c2:c3:c47_10(+(x, 1)) <=> c3(gen_c2:c3:c47_10(x)) gen_c5:c68_10(0) <=> c5 gen_c5:c68_10(+(x, 1)) <=> c6(gen_c5:c68_10(x), c9) gen_c7:c8:c99_10(0) <=> c9 gen_c7:c8:c99_10(+(x, 1)) <=> c7(c, gen_c7:c8:c99_10(x)) The following defined symbols remain to be analysed: G, SEL, activate, ACTIVATE, g, sel They will be analysed ascendingly in the following order: G < ACTIVATE activate < SEL ACTIVATE < SEL activate < ACTIVATE g < activate activate < sel ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: G(gen_0':s:cons:n__f:n__g6_10(n11_10)) -> gen_c2:c3:c47_10(n11_10), rt in Omega(1 + n11_10) Induction Base: G(gen_0':s:cons:n__f:n__g6_10(0)) ->_R^Omega(1) c2 Induction Step: G(gen_0':s:cons:n__f:n__g6_10(+(n11_10, 1))) ->_R^Omega(1) c3(G(gen_0':s:cons:n__f:n__g6_10(n11_10))) ->_IH c3(gen_c2:c3:c47_10(c12_10)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (16) Complex Obligation (BEST) ---------------------------------------- (17) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: F(z0) -> c F(z0) -> c1 G(0') -> c2 G(s(z0)) -> c3(G(z0)) G(z0) -> c4 SEL(0', cons(z0, z1)) -> c5 SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__f(z0)) -> c7(F(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(z0) -> c9 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0') -> s(0') g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) sel(0', cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 Types: F :: 0':s:cons:n__f:n__g -> c:c1 c :: c:c1 c1 :: c:c1 G :: 0':s:cons:n__f:n__g -> c2:c3:c4 0' :: 0':s:cons:n__f:n__g c2 :: c2:c3:c4 s :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c3 :: c2:c3:c4 -> c2:c3:c4 c4 :: c2:c3:c4 SEL :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g -> c5:c6 cons :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c5 :: c5:c6 c6 :: c5:c6 -> c7:c8:c9 -> c5:c6 activate :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g ACTIVATE :: 0':s:cons:n__f:n__g -> c7:c8:c9 n__f :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c7 :: c:c1 -> c7:c8:c9 -> c7:c8:c9 n__g :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c8 :: c2:c3:c4 -> c7:c8:c9 -> c7:c8:c9 c9 :: c7:c8:c9 f :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g g :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g sel :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g hole_c:c11_10 :: c:c1 hole_0':s:cons:n__f:n__g2_10 :: 0':s:cons:n__f:n__g hole_c2:c3:c43_10 :: c2:c3:c4 hole_c5:c64_10 :: c5:c6 hole_c7:c8:c95_10 :: c7:c8:c9 gen_0':s:cons:n__f:n__g6_10 :: Nat -> 0':s:cons:n__f:n__g gen_c2:c3:c47_10 :: Nat -> c2:c3:c4 gen_c5:c68_10 :: Nat -> c5:c6 gen_c7:c8:c99_10 :: Nat -> c7:c8:c9 Generator Equations: gen_0':s:cons:n__f:n__g6_10(0) <=> 0' gen_0':s:cons:n__f:n__g6_10(+(x, 1)) <=> s(gen_0':s:cons:n__f:n__g6_10(x)) gen_c2:c3:c47_10(0) <=> c2 gen_c2:c3:c47_10(+(x, 1)) <=> c3(gen_c2:c3:c47_10(x)) gen_c5:c68_10(0) <=> c5 gen_c5:c68_10(+(x, 1)) <=> c6(gen_c5:c68_10(x), c9) gen_c7:c8:c99_10(0) <=> c9 gen_c7:c8:c99_10(+(x, 1)) <=> c7(c, gen_c7:c8:c99_10(x)) The following defined symbols remain to be analysed: G, SEL, activate, ACTIVATE, g, sel They will be analysed ascendingly in the following order: G < ACTIVATE activate < SEL ACTIVATE < SEL activate < ACTIVATE g < activate activate < sel ---------------------------------------- (18) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (19) BOUNDS(n^1, INF) ---------------------------------------- (20) Obligation: Innermost TRS: Rules: F(z0) -> c F(z0) -> c1 G(0') -> c2 G(s(z0)) -> c3(G(z0)) G(z0) -> c4 SEL(0', cons(z0, z1)) -> c5 SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__f(z0)) -> c7(F(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(z0) -> c9 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0') -> s(0') g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) sel(0', cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 Types: F :: 0':s:cons:n__f:n__g -> c:c1 c :: c:c1 c1 :: c:c1 G :: 0':s:cons:n__f:n__g -> c2:c3:c4 0' :: 0':s:cons:n__f:n__g c2 :: c2:c3:c4 s :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c3 :: c2:c3:c4 -> c2:c3:c4 c4 :: c2:c3:c4 SEL :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g -> c5:c6 cons :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c5 :: c5:c6 c6 :: c5:c6 -> c7:c8:c9 -> c5:c6 activate :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g ACTIVATE :: 0':s:cons:n__f:n__g -> c7:c8:c9 n__f :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c7 :: c:c1 -> c7:c8:c9 -> c7:c8:c9 n__g :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c8 :: c2:c3:c4 -> c7:c8:c9 -> c7:c8:c9 c9 :: c7:c8:c9 f :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g g :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g sel :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g hole_c:c11_10 :: c:c1 hole_0':s:cons:n__f:n__g2_10 :: 0':s:cons:n__f:n__g hole_c2:c3:c43_10 :: c2:c3:c4 hole_c5:c64_10 :: c5:c6 hole_c7:c8:c95_10 :: c7:c8:c9 gen_0':s:cons:n__f:n__g6_10 :: Nat -> 0':s:cons:n__f:n__g gen_c2:c3:c47_10 :: Nat -> c2:c3:c4 gen_c5:c68_10 :: Nat -> c5:c6 gen_c7:c8:c99_10 :: Nat -> c7:c8:c9 Lemmas: G(gen_0':s:cons:n__f:n__g6_10(n11_10)) -> gen_c2:c3:c47_10(n11_10), rt in Omega(1 + n11_10) Generator Equations: gen_0':s:cons:n__f:n__g6_10(0) <=> 0' gen_0':s:cons:n__f:n__g6_10(+(x, 1)) <=> s(gen_0':s:cons:n__f:n__g6_10(x)) gen_c2:c3:c47_10(0) <=> c2 gen_c2:c3:c47_10(+(x, 1)) <=> c3(gen_c2:c3:c47_10(x)) gen_c5:c68_10(0) <=> c5 gen_c5:c68_10(+(x, 1)) <=> c6(gen_c5:c68_10(x), c9) gen_c7:c8:c99_10(0) <=> c9 gen_c7:c8:c99_10(+(x, 1)) <=> c7(c, gen_c7:c8:c99_10(x)) The following defined symbols remain to be analysed: g, SEL, activate, ACTIVATE, sel They will be analysed ascendingly in the following order: activate < SEL ACTIVATE < SEL activate < ACTIVATE g < activate activate < sel ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: g(gen_0':s:cons:n__f:n__g6_10(n364_10)) -> gen_0':s:cons:n__f:n__g6_10(+(1, *(2, n364_10))), rt in Omega(0) Induction Base: g(gen_0':s:cons:n__f:n__g6_10(0)) ->_R^Omega(0) s(0') Induction Step: g(gen_0':s:cons:n__f:n__g6_10(+(n364_10, 1))) ->_R^Omega(0) s(s(g(gen_0':s:cons:n__f:n__g6_10(n364_10)))) ->_IH s(s(gen_0':s:cons:n__f:n__g6_10(+(1, *(2, c365_10))))) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (22) Obligation: Innermost TRS: Rules: F(z0) -> c F(z0) -> c1 G(0') -> c2 G(s(z0)) -> c3(G(z0)) G(z0) -> c4 SEL(0', cons(z0, z1)) -> c5 SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__f(z0)) -> c7(F(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(z0) -> c9 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0') -> s(0') g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) sel(0', cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 Types: F :: 0':s:cons:n__f:n__g -> c:c1 c :: c:c1 c1 :: c:c1 G :: 0':s:cons:n__f:n__g -> c2:c3:c4 0' :: 0':s:cons:n__f:n__g c2 :: c2:c3:c4 s :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c3 :: c2:c3:c4 -> c2:c3:c4 c4 :: c2:c3:c4 SEL :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g -> c5:c6 cons :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c5 :: c5:c6 c6 :: c5:c6 -> c7:c8:c9 -> c5:c6 activate :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g ACTIVATE :: 0':s:cons:n__f:n__g -> c7:c8:c9 n__f :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c7 :: c:c1 -> c7:c8:c9 -> c7:c8:c9 n__g :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g c8 :: c2:c3:c4 -> c7:c8:c9 -> c7:c8:c9 c9 :: c7:c8:c9 f :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g g :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g sel :: 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g -> 0':s:cons:n__f:n__g hole_c:c11_10 :: c:c1 hole_0':s:cons:n__f:n__g2_10 :: 0':s:cons:n__f:n__g hole_c2:c3:c43_10 :: c2:c3:c4 hole_c5:c64_10 :: c5:c6 hole_c7:c8:c95_10 :: c7:c8:c9 gen_0':s:cons:n__f:n__g6_10 :: Nat -> 0':s:cons:n__f:n__g gen_c2:c3:c47_10 :: Nat -> c2:c3:c4 gen_c5:c68_10 :: Nat -> c5:c6 gen_c7:c8:c99_10 :: Nat -> c7:c8:c9 Lemmas: G(gen_0':s:cons:n__f:n__g6_10(n11_10)) -> gen_c2:c3:c47_10(n11_10), rt in Omega(1 + n11_10) g(gen_0':s:cons:n__f:n__g6_10(n364_10)) -> gen_0':s:cons:n__f:n__g6_10(+(1, *(2, n364_10))), rt in Omega(0) Generator Equations: gen_0':s:cons:n__f:n__g6_10(0) <=> 0' gen_0':s:cons:n__f:n__g6_10(+(x, 1)) <=> s(gen_0':s:cons:n__f:n__g6_10(x)) gen_c2:c3:c47_10(0) <=> c2 gen_c2:c3:c47_10(+(x, 1)) <=> c3(gen_c2:c3:c47_10(x)) gen_c5:c68_10(0) <=> c5 gen_c5:c68_10(+(x, 1)) <=> c6(gen_c5:c68_10(x), c9) gen_c7:c8:c99_10(0) <=> c9 gen_c7:c8:c99_10(+(x, 1)) <=> c7(c, gen_c7:c8:c99_10(x)) The following defined symbols remain to be analysed: activate, SEL, ACTIVATE, sel They will be analysed ascendingly in the following order: activate < SEL ACTIVATE < SEL activate < ACTIVATE activate < sel ---------------------------------------- (23) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 Tuples: F(z0) -> c F(z0) -> c1 G(0) -> c2 G(s(z0)) -> c3(G(z0)) G(z0) -> c4 SEL(0, cons(z0, z1)) -> c5 SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__f(z0)) -> c7(F(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(z0) -> c9 S tuples: F(z0) -> c F(z0) -> c1 G(0) -> c2 G(s(z0)) -> c3(G(z0)) G(z0) -> c4 SEL(0, cons(z0, z1)) -> c5 SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__f(z0)) -> c7(F(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(z0) -> c9 K tuples:none Defined Rule Symbols: f_1, g_1, sel_2, activate_1 Defined Pair Symbols: F_1, G_1, SEL_2, ACTIVATE_1 Compound Symbols: c, c1, c2, c3_1, c4, c5, c6_2, c7_2, c8_2, c9 ---------------------------------------- (25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: G(z0) -> c4 F(z0) -> c ACTIVATE(z0) -> c9 SEL(0, cons(z0, z1)) -> c5 G(0) -> c2 F(z0) -> c1 ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 Tuples: G(s(z0)) -> c3(G(z0)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__f(z0)) -> c7(F(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) S tuples: G(s(z0)) -> c3(G(z0)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__f(z0)) -> c7(F(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) K tuples:none Defined Rule Symbols: f_1, g_1, sel_2, activate_1 Defined Pair Symbols: G_1, SEL_2, ACTIVATE_1 Compound Symbols: c3_1, c6_2, c7_2, c8_2 ---------------------------------------- (27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 Tuples: G(s(z0)) -> c3(G(z0)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) S tuples: G(s(z0)) -> c3(G(z0)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) K tuples:none Defined Rule Symbols: f_1, g_1, sel_2, activate_1 Defined Pair Symbols: G_1, SEL_2, ACTIVATE_1 Compound Symbols: c3_1, c6_2, c8_2, c7_1 ---------------------------------------- (29) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: G(s(z0)) -> c3(G(z0)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) S tuples: G(s(z0)) -> c3(G(z0)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) K tuples:none Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: G_1, SEL_2, ACTIVATE_1 Compound Symbols: c3_1, c6_2, c8_2, c7_1 ---------------------------------------- (31) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (32) 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: G(s(z0)) -> c3(G(z0)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) The (relative) TRS S consists of the following rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Rewrite Strategy: INNERMOST ---------------------------------------- (33) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (34) 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: G(s(z0)) -> c3(G(z0)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (35) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (36) 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: G(s(z0)) -> c3(G(z0)) [1] SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) [1] ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) [1] ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) [1] activate(n__f(z0)) -> f(activate(z0)) [0] activate(n__g(z0)) -> g(activate(z0)) [0] activate(z0) -> z0 [0] f(z0) -> cons(z0, n__f(n__g(z0))) [0] f(z0) -> n__f(z0) [0] g(0) -> s(0) [0] g(s(z0)) -> s(s(g(z0))) [0] g(z0) -> n__g(z0) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (37) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (38) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: G(s(z0)) -> c3(G(z0)) [1] SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) [1] ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) [1] ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) [1] activate(n__f(z0)) -> f(activate(z0)) [0] activate(n__g(z0)) -> g(activate(z0)) [0] activate(z0) -> z0 [0] f(z0) -> cons(z0, n__f(n__g(z0))) [0] f(z0) -> n__f(z0) [0] g(0) -> s(0) [0] g(s(z0)) -> s(s(g(z0))) [0] g(z0) -> n__g(z0) [0] The TRS has the following type information: G :: s:cons:n__g:n__f:0 -> c3 s :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 c3 :: c3 -> c3 SEL :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 -> c6 cons :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 c6 :: c6 -> c8:c7 -> c6 activate :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 ACTIVATE :: s:cons:n__g:n__f:0 -> c8:c7 n__g :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 c8 :: c3 -> c8:c7 -> c8:c7 n__f :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 c7 :: c8:c7 -> c8:c7 f :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 g :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 0 :: s:cons:n__g:n__f:0 Rewrite Strategy: INNERMOST ---------------------------------------- (39) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: G_1 SEL_2 ACTIVATE_1 (c) The following functions are completely defined: activate_1 f_1 g_1 Due to the following rules being added: activate(v0) -> 0 [0] f(v0) -> 0 [0] g(v0) -> 0 [0] And the following fresh constants: const, const1, const2 ---------------------------------------- (40) 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: G(s(z0)) -> c3(G(z0)) [1] SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) [1] ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) [1] ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) [1] activate(n__f(z0)) -> f(activate(z0)) [0] activate(n__g(z0)) -> g(activate(z0)) [0] activate(z0) -> z0 [0] f(z0) -> cons(z0, n__f(n__g(z0))) [0] f(z0) -> n__f(z0) [0] g(0) -> s(0) [0] g(s(z0)) -> s(s(g(z0))) [0] g(z0) -> n__g(z0) [0] activate(v0) -> 0 [0] f(v0) -> 0 [0] g(v0) -> 0 [0] The TRS has the following type information: G :: s:cons:n__g:n__f:0 -> c3 s :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 c3 :: c3 -> c3 SEL :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 -> c6 cons :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 c6 :: c6 -> c8:c7 -> c6 activate :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 ACTIVATE :: s:cons:n__g:n__f:0 -> c8:c7 n__g :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 c8 :: c3 -> c8:c7 -> c8:c7 n__f :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 c7 :: c8:c7 -> c8:c7 f :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 g :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 0 :: s:cons:n__g:n__f:0 const :: c3 const1 :: c6 const2 :: c8:c7 Rewrite Strategy: INNERMOST ---------------------------------------- (41) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (42) 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: G(s(z0)) -> c3(G(z0)) [1] SEL(s(z0), cons(z1, n__f(z0'))) -> c6(SEL(z0, f(activate(z0'))), ACTIVATE(n__f(z0'))) [1] SEL(s(z0), cons(z1, n__g(z0''))) -> c6(SEL(z0, g(activate(z0''))), ACTIVATE(n__g(z0''))) [1] SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2), ACTIVATE(z2)) [1] SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, 0), ACTIVATE(z2)) [1] ACTIVATE(n__g(n__f(z01))) -> c8(G(f(activate(z01))), ACTIVATE(n__f(z01))) [1] ACTIVATE(n__g(n__g(z02))) -> c8(G(g(activate(z02))), ACTIVATE(n__g(z02))) [1] ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) [1] ACTIVATE(n__g(z0)) -> c8(G(0), ACTIVATE(z0)) [1] ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) [1] activate(n__f(n__f(z03))) -> f(f(activate(z03))) [0] activate(n__f(n__g(z04))) -> f(g(activate(z04))) [0] activate(n__f(z0)) -> f(z0) [0] activate(n__f(z0)) -> f(0) [0] activate(n__g(n__f(z05))) -> g(f(activate(z05))) [0] activate(n__g(n__g(z06))) -> g(g(activate(z06))) [0] activate(n__g(z0)) -> g(z0) [0] activate(n__g(z0)) -> g(0) [0] activate(z0) -> z0 [0] f(z0) -> cons(z0, n__f(n__g(z0))) [0] f(z0) -> n__f(z0) [0] g(0) -> s(0) [0] g(s(z0)) -> s(s(g(z0))) [0] g(z0) -> n__g(z0) [0] activate(v0) -> 0 [0] f(v0) -> 0 [0] g(v0) -> 0 [0] The TRS has the following type information: G :: s:cons:n__g:n__f:0 -> c3 s :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 c3 :: c3 -> c3 SEL :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 -> c6 cons :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 c6 :: c6 -> c8:c7 -> c6 activate :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 ACTIVATE :: s:cons:n__g:n__f:0 -> c8:c7 n__g :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 c8 :: c3 -> c8:c7 -> c8:c7 n__f :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 c7 :: c8:c7 -> c8:c7 f :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 g :: s:cons:n__g:n__f:0 -> s:cons:n__g:n__f:0 0 :: s:cons:n__g:n__f:0 const :: c3 const1 :: c6 const2 :: c8:c7 Rewrite Strategy: INNERMOST ---------------------------------------- (43) 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 const1 => 0 const2 => 0 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z0) :|: z = 1 + z0, z0 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(z0) + ACTIVATE(z0) :|: z = 1 + z0, z0 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z02))) + ACTIVATE(1 + z02) :|: z = 1 + (1 + z02), z02 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z01))) + ACTIVATE(1 + z01) :|: z = 1 + (1 + z01), z01 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(0) + ACTIVATE(z0) :|: z = 1 + z0, z0 >= 0 G(z) -{ 1 }-> 1 + G(z0) :|: z = 1 + z0, z0 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z0, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z0, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z = 1 + z0, z0 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z0, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z = 1 + z0, z0 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z0, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0 activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0 activate(z) -{ 0 }-> g(z0) :|: z = 1 + z0, z0 >= 0 activate(z) -{ 0 }-> g(g(activate(z06))) :|: z06 >= 0, z = 1 + (1 + z06) activate(z) -{ 0 }-> g(f(activate(z05))) :|: z05 >= 0, z = 1 + (1 + z05) activate(z) -{ 0 }-> g(0) :|: z = 1 + z0, z0 >= 0 activate(z) -{ 0 }-> f(z0) :|: z = 1 + z0, z0 >= 0 activate(z) -{ 0 }-> f(g(activate(z04))) :|: z04 >= 0, z = 1 + (1 + z04) activate(z) -{ 0 }-> f(f(activate(z03))) :|: z = 1 + (1 + z03), z03 >= 0 activate(z) -{ 0 }-> f(0) :|: z = 1 + z0, z0 >= 0 activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 f(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 f(z) -{ 0 }-> 1 + z0 + (1 + (1 + z0)) :|: z = z0, z0 >= 0 g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 g(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + g(z0)) :|: z = 1 + z0, z0 >= 0 ---------------------------------------- (45) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: f(z) -{ 0 }-> 1 + z0 + (1 + (1 + z0)) :|: z = z0, z0 >= 0 f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 f(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z0) :|: z = 1 + z0, z0 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(z0) + ACTIVATE(z0) :|: z = 1 + z0, z0 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z02))) + ACTIVATE(1 + z02) :|: z = 1 + (1 + z02), z02 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z01))) + ACTIVATE(1 + z01) :|: z = 1 + (1 + z01), z01 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(0) + ACTIVATE(z0) :|: z = 1 + z0, z0 >= 0 G(z) -{ 1 }-> 1 + G(z0) :|: z = 1 + z0, z0 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z0, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z0, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z = 1 + z0, z0 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z0, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z = 1 + z0, z0 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z0, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0 activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0 activate(z) -{ 0 }-> g(z0) :|: z = 1 + z0, z0 >= 0 activate(z) -{ 0 }-> g(g(activate(z06))) :|: z06 >= 0, z = 1 + (1 + z06) activate(z) -{ 0 }-> g(f(activate(z05))) :|: z05 >= 0, z = 1 + (1 + z05) activate(z) -{ 0 }-> g(0) :|: z = 1 + z0, z0 >= 0 activate(z) -{ 0 }-> f(g(activate(z04))) :|: z04 >= 0, z = 1 + (1 + z04) activate(z) -{ 0 }-> f(f(activate(z03))) :|: z = 1 + (1 + z03), z03 >= 0 activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 activate(z) -{ 0 }-> 0 :|: z = 1 + z0, z0 >= 0, v0 >= 0, z0 = v0 activate(z) -{ 0 }-> 0 :|: z = 1 + z0, z0 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z = 1 + z0, z0 >= 0, z0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z = 1 + z0, z0 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z = 1 + z0, z0 >= 0, z0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z = 1 + z0, z0 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 f(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 f(z) -{ 0 }-> 1 + z0 + (1 + (1 + z0)) :|: z = z0, z0 >= 0 g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 g(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + g(z0)) :|: z = 1 + z0, z0 >= 0 ---------------------------------------- (47) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(0) + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(z - 1) + ACTIVATE(z - 1) :|: z - 1 >= 0 G(z) -{ 1 }-> 1 + G(z - 1) :|: z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(0) :|: z - 1 >= 0 activate(z) -{ 0 }-> g(z - 1) :|: z - 1 >= 0 activate(z) -{ 0 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + g(z - 1)) :|: z - 1 >= 0 ---------------------------------------- (49) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { g } { f } { G } { activate } { ACTIVATE } { SEL } ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(0) + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(z - 1) + ACTIVATE(z - 1) :|: z - 1 >= 0 G(z) -{ 1 }-> 1 + G(z - 1) :|: z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(0) :|: z - 1 >= 0 activate(z) -{ 0 }-> g(z - 1) :|: z - 1 >= 0 activate(z) -{ 0 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + g(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {g}, {f}, {G}, {activate}, {ACTIVATE}, {SEL} ---------------------------------------- (51) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(0) + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(z - 1) + ACTIVATE(z - 1) :|: z - 1 >= 0 G(z) -{ 1 }-> 1 + G(z - 1) :|: z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(0) :|: z - 1 >= 0 activate(z) -{ 0 }-> g(z - 1) :|: z - 1 >= 0 activate(z) -{ 0 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + g(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {g}, {f}, {G}, {activate}, {ACTIVATE}, {SEL} ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 2*z ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(0) + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(z - 1) + ACTIVATE(z - 1) :|: z - 1 >= 0 G(z) -{ 1 }-> 1 + G(z - 1) :|: z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(0) :|: z - 1 >= 0 activate(z) -{ 0 }-> g(z - 1) :|: z - 1 >= 0 activate(z) -{ 0 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + g(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {g}, {f}, {G}, {activate}, {ACTIVATE}, {SEL} Previous analysis results are: g: runtime: ?, size: O(n^1) [1 + 2*z] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(0) + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(z - 1) + ACTIVATE(z - 1) :|: z - 1 >= 0 G(z) -{ 1 }-> 1 + G(z - 1) :|: z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(0) :|: z - 1 >= 0 activate(z) -{ 0 }-> g(z - 1) :|: z - 1 >= 0 activate(z) -{ 0 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + g(z - 1)) :|: z - 1 >= 0 Function symbols to be analyzed: {f}, {G}, {activate}, {ACTIVATE}, {SEL} Previous analysis results are: g: runtime: O(1) [0], size: O(n^1) [1 + 2*z] ---------------------------------------- (57) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(0) + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(z - 1) + ACTIVATE(z - 1) :|: z - 1 >= 0 G(z) -{ 1 }-> 1 + G(z - 1) :|: z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> s :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 0 }-> s' :|: s' >= 0, s' <= 2 * 0 + 1, z - 1 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1) + 1, z - 1 >= 0 Function symbols to be analyzed: {f}, {G}, {activate}, {ACTIVATE}, {SEL} Previous analysis results are: g: runtime: O(1) [0], size: O(n^1) [1 + 2*z] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 2*z ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(0) + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(z - 1) + ACTIVATE(z - 1) :|: z - 1 >= 0 G(z) -{ 1 }-> 1 + G(z - 1) :|: z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> s :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 0 }-> s' :|: s' >= 0, s' <= 2 * 0 + 1, z - 1 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1) + 1, z - 1 >= 0 Function symbols to be analyzed: {f}, {G}, {activate}, {ACTIVATE}, {SEL} Previous analysis results are: g: runtime: O(1) [0], size: O(n^1) [1 + 2*z] f: runtime: ?, size: O(n^1) [3 + 2*z] ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(0) + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(z - 1) + ACTIVATE(z - 1) :|: z - 1 >= 0 G(z) -{ 1 }-> 1 + G(z - 1) :|: z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> s :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 0 }-> s' :|: s' >= 0, s' <= 2 * 0 + 1, z - 1 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1) + 1, z - 1 >= 0 Function symbols to be analyzed: {G}, {activate}, {ACTIVATE}, {SEL} Previous analysis results are: g: runtime: O(1) [0], size: O(n^1) [1 + 2*z] f: runtime: O(1) [0], size: O(n^1) [3 + 2*z] ---------------------------------------- (63) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(0) + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(z - 1) + ACTIVATE(z - 1) :|: z - 1 >= 0 G(z) -{ 1 }-> 1 + G(z - 1) :|: z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> s :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 0 }-> s' :|: s' >= 0, s' <= 2 * 0 + 1, z - 1 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1) + 1, z - 1 >= 0 Function symbols to be analyzed: {G}, {activate}, {ACTIVATE}, {SEL} Previous analysis results are: g: runtime: O(1) [0], size: O(n^1) [1 + 2*z] f: runtime: O(1) [0], size: O(n^1) [3 + 2*z] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: G after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(0) + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(z - 1) + ACTIVATE(z - 1) :|: z - 1 >= 0 G(z) -{ 1 }-> 1 + G(z - 1) :|: z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> s :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 0 }-> s' :|: s' >= 0, s' <= 2 * 0 + 1, z - 1 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1) + 1, z - 1 >= 0 Function symbols to be analyzed: {G}, {activate}, {ACTIVATE}, {SEL} Previous analysis results are: g: runtime: O(1) [0], size: O(n^1) [1 + 2*z] f: runtime: O(1) [0], size: O(n^1) [3 + 2*z] G: runtime: ?, size: O(1) [0] ---------------------------------------- (67) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: G after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(0) + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(z - 1) + ACTIVATE(z - 1) :|: z - 1 >= 0 G(z) -{ 1 }-> 1 + G(z - 1) :|: z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> s :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 0 }-> s' :|: s' >= 0, s' <= 2 * 0 + 1, z - 1 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1) + 1, z - 1 >= 0 Function symbols to be analyzed: {activate}, {ACTIVATE}, {SEL} Previous analysis results are: g: runtime: O(1) [0], size: O(n^1) [1 + 2*z] f: runtime: O(1) [0], size: O(n^1) [3 + 2*z] G: runtime: O(n^1) [z], size: O(1) [0] ---------------------------------------- (69) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ z }-> 1 + s2 + ACTIVATE(z - 1) :|: s2 >= 0, s2 <= 0, z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + s3 + ACTIVATE(z - 1) :|: s3 >= 0, s3 <= 0, z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 G(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> s :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 0 }-> s' :|: s' >= 0, s' <= 2 * 0 + 1, z - 1 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1) + 1, z - 1 >= 0 Function symbols to be analyzed: {activate}, {ACTIVATE}, {SEL} Previous analysis results are: g: runtime: O(1) [0], size: O(n^1) [1 + 2*z] f: runtime: O(1) [0], size: O(n^1) [3 + 2*z] G: runtime: O(n^1) [z], size: O(1) [0] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: activate after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ z }-> 1 + s2 + ACTIVATE(z - 1) :|: s2 >= 0, s2 <= 0, z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + s3 + ACTIVATE(z - 1) :|: s3 >= 0, s3 <= 0, z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 G(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> s :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 0 }-> s' :|: s' >= 0, s' <= 2 * 0 + 1, z - 1 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1) + 1, z - 1 >= 0 Function symbols to be analyzed: {activate}, {ACTIVATE}, {SEL} Previous analysis results are: g: runtime: O(1) [0], size: O(n^1) [1 + 2*z] f: runtime: O(1) [0], size: O(n^1) [3 + 2*z] G: runtime: O(n^1) [z], size: O(1) [0] activate: runtime: ?, size: EXP ---------------------------------------- (73) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: activate after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ z }-> 1 + s2 + ACTIVATE(z - 1) :|: s2 >= 0, s2 <= 0, z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + s3 + ACTIVATE(z - 1) :|: s3 >= 0, s3 <= 0, z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(g(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(f(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0 G(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, g(activate(z0''))) + ACTIVATE(1 + z0'') :|: z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, f(activate(z0'))) + ACTIVATE(1 + z0') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> s :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 0 }-> s' :|: s' >= 0, s' <= 2 * 0 + 1, z - 1 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1) + 1, z - 1 >= 0 Function symbols to be analyzed: {ACTIVATE}, {SEL} Previous analysis results are: g: runtime: O(1) [0], size: O(n^1) [1 + 2*z] f: runtime: O(1) [0], size: O(n^1) [3 + 2*z] G: runtime: O(n^1) [z], size: O(1) [0] activate: runtime: O(1) [0], size: EXP ---------------------------------------- (75) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 + s15 }-> 1 + s16 + ACTIVATE(1 + (z - 2)) :|: s14 >= 0, s14 <= inf2, s15 >= 0, s15 <= 2 * s14 + 3, s16 >= 0, s16 <= 0, z - 2 >= 0 ACTIVATE(z) -{ z }-> 1 + s2 + ACTIVATE(z - 1) :|: s2 >= 0, s2 <= 0, z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + s3 + ACTIVATE(z - 1) :|: s3 >= 0, s3 <= 0, z - 1 >= 0 ACTIVATE(z) -{ 1 + s7 }-> 1 + s8 + ACTIVATE(1 + (z - 2)) :|: s6 >= 0, s6 <= inf', s7 >= 0, s7 <= 2 * s6 + 1, s8 >= 0, s8 <= 0, z - 2 >= 0 G(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, s13) + ACTIVATE(1 + z0') :|: s12 >= 0, s12 <= inf1, s13 >= 0, s13 <= 2 * s12 + 3, z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, s5) + ACTIVATE(1 + z0'') :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= 2 * s4 + 1, z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> s :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 0 }-> s' :|: s' >= 0, s' <= 2 * 0 + 1, z - 1 >= 0 activate(z) -{ 0 }-> s11 :|: s9 >= 0, s9 <= inf'', s10 >= 0, s10 <= 2 * s9 + 1, s11 >= 0, s11 <= 2 * s10 + 1, z - 2 >= 0 activate(z) -{ 0 }-> s19 :|: s17 >= 0, s17 <= inf3, s18 >= 0, s18 <= 2 * s17 + 3, s19 >= 0, s19 <= 2 * s18 + 3, z - 2 >= 0 activate(z) -{ 0 }-> s22 :|: s20 >= 0, s20 <= inf4, s21 >= 0, s21 <= 2 * s20 + 1, s22 >= 0, s22 <= 2 * s21 + 3, z - 2 >= 0 activate(z) -{ 0 }-> s25 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= 2 * s23 + 3, s25 >= 0, s25 <= 2 * s24 + 1, z - 2 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1) + 1, z - 1 >= 0 Function symbols to be analyzed: {ACTIVATE}, {SEL} Previous analysis results are: g: runtime: O(1) [0], size: O(n^1) [1 + 2*z] f: runtime: O(1) [0], size: O(n^1) [3 + 2*z] G: runtime: O(n^1) [z], size: O(1) [0] activate: runtime: O(1) [0], size: EXP ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: ACTIVATE after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 + s15 }-> 1 + s16 + ACTIVATE(1 + (z - 2)) :|: s14 >= 0, s14 <= inf2, s15 >= 0, s15 <= 2 * s14 + 3, s16 >= 0, s16 <= 0, z - 2 >= 0 ACTIVATE(z) -{ z }-> 1 + s2 + ACTIVATE(z - 1) :|: s2 >= 0, s2 <= 0, z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + s3 + ACTIVATE(z - 1) :|: s3 >= 0, s3 <= 0, z - 1 >= 0 ACTIVATE(z) -{ 1 + s7 }-> 1 + s8 + ACTIVATE(1 + (z - 2)) :|: s6 >= 0, s6 <= inf', s7 >= 0, s7 <= 2 * s6 + 1, s8 >= 0, s8 <= 0, z - 2 >= 0 G(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, s13) + ACTIVATE(1 + z0') :|: s12 >= 0, s12 <= inf1, s13 >= 0, s13 <= 2 * s12 + 3, z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, s5) + ACTIVATE(1 + z0'') :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= 2 * s4 + 1, z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> s :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 0 }-> s' :|: s' >= 0, s' <= 2 * 0 + 1, z - 1 >= 0 activate(z) -{ 0 }-> s11 :|: s9 >= 0, s9 <= inf'', s10 >= 0, s10 <= 2 * s9 + 1, s11 >= 0, s11 <= 2 * s10 + 1, z - 2 >= 0 activate(z) -{ 0 }-> s19 :|: s17 >= 0, s17 <= inf3, s18 >= 0, s18 <= 2 * s17 + 3, s19 >= 0, s19 <= 2 * s18 + 3, z - 2 >= 0 activate(z) -{ 0 }-> s22 :|: s20 >= 0, s20 <= inf4, s21 >= 0, s21 <= 2 * s20 + 1, s22 >= 0, s22 <= 2 * s21 + 3, z - 2 >= 0 activate(z) -{ 0 }-> s25 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= 2 * s23 + 3, s25 >= 0, s25 <= 2 * s24 + 1, z - 2 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1) + 1, z - 1 >= 0 Function symbols to be analyzed: {ACTIVATE}, {SEL} Previous analysis results are: g: runtime: O(1) [0], size: O(n^1) [1 + 2*z] f: runtime: O(1) [0], size: O(n^1) [3 + 2*z] G: runtime: O(n^1) [z], size: O(1) [0] activate: runtime: O(1) [0], size: EXP ACTIVATE: runtime: ?, size: O(1) [0] ---------------------------------------- (79) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: ACTIVATE after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (80) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0 ACTIVATE(z) -{ 1 + s15 }-> 1 + s16 + ACTIVATE(1 + (z - 2)) :|: s14 >= 0, s14 <= inf2, s15 >= 0, s15 <= 2 * s14 + 3, s16 >= 0, s16 <= 0, z - 2 >= 0 ACTIVATE(z) -{ z }-> 1 + s2 + ACTIVATE(z - 1) :|: s2 >= 0, s2 <= 0, z - 1 >= 0 ACTIVATE(z) -{ 1 }-> 1 + s3 + ACTIVATE(z - 1) :|: s3 >= 0, s3 <= 0, z - 1 >= 0 ACTIVATE(z) -{ 1 + s7 }-> 1 + s8 + ACTIVATE(1 + (z - 2)) :|: s6 >= 0, s6 <= inf', s7 >= 0, s7 <= 2 * s6 + 1, s8 >= 0, s8 <= 0, z - 2 >= 0 G(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, s13) + ACTIVATE(1 + z0') :|: s12 >= 0, s12 <= inf1, s13 >= 0, s13 <= 2 * s12 + 3, z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, s5) + ACTIVATE(1 + z0'') :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= 2 * s4 + 1, z1 >= 0, z' = 1 + z1 + (1 + z0''), z - 1 >= 0, z0'' >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, z2) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 SEL(z, z') -{ 1 }-> 1 + SEL(z - 1, 0) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0 activate(z) -{ 0 }-> s :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 0 }-> s' :|: s' >= 0, s' <= 2 * 0 + 1, z - 1 >= 0 activate(z) -{ 0 }-> s11 :|: s9 >= 0, s9 <= inf'', s10 >= 0, s10 <= 2 * s9 + 1, s11 >= 0, s11 <= 2 * s10 + 1, z - 2 >= 0 activate(z) -{ 0 }-> s19 :|: s17 >= 0, s17 <= inf3, s18 >= 0, s18 <= 2 * s17 + 3, s19 >= 0, s19 <= 2 * s18 + 3, z - 2 >= 0 activate(z) -{ 0 }-> s22 :|: s20 >= 0, s20 <= inf4, s21 >= 0, s21 <= 2 * s20 + 1, s22 >= 0, s22 <= 2 * s21 + 3, z - 2 >= 0 activate(z) -{ 0 }-> s25 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= 2 * s23 + 3, s25 >= 0, s25 <= 2 * s24 + 1, z - 2 >= 0 activate(z) -{ 0 }-> z :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z >= 0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0 activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, 0 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0 activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, 0 = z0', z0' >= 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 0 }-> 1 + z :|: z >= 0 f(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 0 }-> 0 :|: z >= 0 g(z) -{ 0 }-> 1 + z :|: z >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1) + 1, z - 1 >= 0 Function symbols to be analyzed: {ACTIVATE}, {SEL} Previous analysis results are: g: runtime: O(1) [0], size: O(n^1) [1 + 2*z] f: runtime: O(1) [0], size: O(n^1) [3 + 2*z] G: runtime: O(n^1) [z], size: O(1) [0] activate: runtime: O(1) [0], size: EXP ACTIVATE: runtime: INF, size: O(1) [0] ---------------------------------------- (81) 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: activate(v0) -> null_activate [0] f(v0) -> null_f [0] g(v0) -> null_g [0] G(v0) -> null_G [0] SEL(v0, v1) -> null_SEL [0] ACTIVATE(v0) -> null_ACTIVATE [0] And the following fresh constants: null_activate, null_f, null_g, null_G, null_SEL, null_ACTIVATE ---------------------------------------- (82) 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: G(s(z0)) -> c3(G(z0)) [1] SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) [1] ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) [1] ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) [1] activate(n__f(z0)) -> f(activate(z0)) [0] activate(n__g(z0)) -> g(activate(z0)) [0] activate(z0) -> z0 [0] f(z0) -> cons(z0, n__f(n__g(z0))) [0] f(z0) -> n__f(z0) [0] g(0) -> s(0) [0] g(s(z0)) -> s(s(g(z0))) [0] g(z0) -> n__g(z0) [0] activate(v0) -> null_activate [0] f(v0) -> null_f [0] g(v0) -> null_g [0] G(v0) -> null_G [0] SEL(v0, v1) -> null_SEL [0] ACTIVATE(v0) -> null_ACTIVATE [0] The TRS has the following type information: G :: s:cons:n__g:n__f:0:null_activate:null_f:null_g -> c3:null_G s :: s:cons:n__g:n__f:0:null_activate:null_f:null_g -> s:cons:n__g:n__f:0:null_activate:null_f:null_g c3 :: c3:null_G -> c3:null_G SEL :: s:cons:n__g:n__f:0:null_activate:null_f:null_g -> s:cons:n__g:n__f:0:null_activate:null_f:null_g -> c6:null_SEL cons :: s:cons:n__g:n__f:0:null_activate:null_f:null_g -> s:cons:n__g:n__f:0:null_activate:null_f:null_g -> s:cons:n__g:n__f:0:null_activate:null_f:null_g c6 :: c6:null_SEL -> c8:c7:null_ACTIVATE -> c6:null_SEL activate :: s:cons:n__g:n__f:0:null_activate:null_f:null_g -> s:cons:n__g:n__f:0:null_activate:null_f:null_g ACTIVATE :: s:cons:n__g:n__f:0:null_activate:null_f:null_g -> c8:c7:null_ACTIVATE n__g :: s:cons:n__g:n__f:0:null_activate:null_f:null_g -> s:cons:n__g:n__f:0:null_activate:null_f:null_g c8 :: c3:null_G -> c8:c7:null_ACTIVATE -> c8:c7:null_ACTIVATE n__f :: s:cons:n__g:n__f:0:null_activate:null_f:null_g -> s:cons:n__g:n__f:0:null_activate:null_f:null_g c7 :: c8:c7:null_ACTIVATE -> c8:c7:null_ACTIVATE f :: s:cons:n__g:n__f:0:null_activate:null_f:null_g -> s:cons:n__g:n__f:0:null_activate:null_f:null_g g :: s:cons:n__g:n__f:0:null_activate:null_f:null_g -> s:cons:n__g:n__f:0:null_activate:null_f:null_g 0 :: s:cons:n__g:n__f:0:null_activate:null_f:null_g null_activate :: s:cons:n__g:n__f:0:null_activate:null_f:null_g null_f :: s:cons:n__g:n__f:0:null_activate:null_f:null_g null_g :: s:cons:n__g:n__f:0:null_activate:null_f:null_g null_G :: c3:null_G null_SEL :: c6:null_SEL null_ACTIVATE :: c8:c7:null_ACTIVATE Rewrite Strategy: INNERMOST ---------------------------------------- (83) 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_activate => 0 null_f => 0 null_g => 0 null_G => 0 null_SEL => 0 null_ACTIVATE => 0 ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: ACTIVATE(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ACTIVATE(z) -{ 1 }-> 1 + ACTIVATE(z0) :|: z = 1 + z0, z0 >= 0 ACTIVATE(z) -{ 1 }-> 1 + G(activate(z0)) + ACTIVATE(z0) :|: z = 1 + z0, z0 >= 0 G(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 G(z) -{ 1 }-> 1 + G(z0) :|: z = 1 + z0, z0 >= 0 SEL(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 SEL(z, z') -{ 1 }-> 1 + SEL(z0, activate(z2)) + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0 activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0 activate(z) -{ 0 }-> g(activate(z0)) :|: z = 1 + z0, z0 >= 0 activate(z) -{ 0 }-> f(activate(z0)) :|: z = 1 + z0, z0 >= 0 activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 f(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 f(z) -{ 0 }-> 1 + z0 + (1 + (1 + z0)) :|: z = z0, z0 >= 0 g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 g(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 g(z) -{ 0 }-> 1 + 0 :|: z = 0 g(z) -{ 0 }-> 1 + (1 + g(z0)) :|: z = 1 + z0, z0 >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) We considered the (Usable) Rules:none And the Tuples: G(s(z0)) -> c3(G(z0)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ACTIVATE(x_1)) = 0 POL(G(x_1)) = 0 POL(SEL(x_1, x_2)) = x_1 POL(activate(x_1)) = [1] POL(c3(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c7(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = [1] + x_1 + x_2 POL(f(x_1)) = [1] + x_1 POL(g(x_1)) = [1] + x_1 POL(n__f(x_1)) = [1] + x_1 POL(n__g(x_1)) = [1] + x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: G(s(z0)) -> c3(G(z0)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) S tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: G_1, SEL_2, ACTIVATE_1 Compound Symbols: c3_1, c6_2, c8_2, c7_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) by SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(activate(z0))), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, g(activate(z0))), ACTIVATE(n__g(z0))) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(activate(z0))), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, g(activate(z0))), ACTIVATE(n__g(z0))) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) S tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: G_1, ACTIVATE_1, SEL_2 Compound Symbols: c3_1, c8_2, c7_1, c6_2 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACTIVATE(n__g(z0)) -> c8(G(activate(z0)), ACTIVATE(z0)) by ACTIVATE(n__g(n__f(z0))) -> c8(G(f(activate(z0))), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(activate(z0))), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, g(activate(z0))), ACTIVATE(n__g(z0))) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(activate(z0))), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) S tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(activate(z0))), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: G_1, ACTIVATE_1, SEL_2 Compound Symbols: c3_1, c7_1, c6_2, c8_2 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(activate(z0))), ACTIVATE(n__f(z0))) by SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, n__f(activate(x2))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, g(activate(z0))), ACTIVATE(n__g(z0))) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(activate(z0))), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, n__f(activate(x2))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) S tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(activate(z0))), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: G_1, ACTIVATE_1, SEL_2 Compound Symbols: c3_1, c7_1, c6_2, c8_2 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, g(activate(z0))), ACTIVATE(n__g(z0))) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(activate(z0))), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) S tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(activate(z0))), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: G_1, ACTIVATE_1, SEL_2 Compound Symbols: c3_1, c7_1, c6_2, c8_2, c6_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, g(activate(z0))), ACTIVATE(n__g(z0))) by SEL(s(x0), cons(x1, n__g(x2))) -> c6(SEL(x0, n__g(activate(x2))), ACTIVATE(n__g(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, g(z0)), ACTIVATE(n__g(z0))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(activate(z0))), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(SEL(x0, n__g(activate(x2))), ACTIVATE(n__g(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, g(z0)), ACTIVATE(n__g(z0))) S tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(activate(z0))), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: G_1, ACTIVATE_1, SEL_2 Compound Symbols: c3_1, c7_1, c6_2, c8_2, c6_1 ---------------------------------------- (97) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(activate(z0))), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, g(z0)), ACTIVATE(n__g(z0))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) S tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(activate(z0))), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: G_1, ACTIVATE_1, SEL_2 Compound Symbols: c3_1, c7_1, c6_2, c8_2, c6_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACTIVATE(n__g(n__f(z0))) -> c8(G(f(activate(z0))), ACTIVATE(n__f(z0))) by ACTIVATE(n__g(n__f(x0))) -> c8(G(cons(activate(x0), n__f(n__g(activate(x0))))), ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__f(x0))) -> c8(G(n__f(activate(x0))), ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, g(z0)), ACTIVATE(n__g(z0))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(x0))) -> c8(G(cons(activate(x0), n__f(n__g(activate(x0))))), ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__f(x0))) -> c8(G(n__f(activate(x0))), ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) S tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) ACTIVATE(n__g(n__f(x0))) -> c8(G(cons(activate(x0), n__f(n__g(activate(x0))))), ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__f(x0))) -> c8(G(n__f(activate(x0))), ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: G_1, ACTIVATE_1, SEL_2 Compound Symbols: c3_1, c7_1, c6_2, c8_2, c6_1 ---------------------------------------- (101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, g(z0)), ACTIVATE(n__g(z0))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) S tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: G_1, ACTIVATE_1, SEL_2 Compound Symbols: c3_1, c7_1, c6_2, c8_2, c6_1, c8_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACTIVATE(n__g(n__g(z0))) -> c8(G(g(activate(z0))), ACTIVATE(n__g(z0))) by ACTIVATE(n__g(n__g(x0))) -> c8(G(n__g(activate(x0))), ACTIVATE(n__g(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, g(z0)), ACTIVATE(n__g(z0))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(x0))) -> c8(G(n__g(activate(x0))), ACTIVATE(n__g(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) S tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(x0))) -> c8(G(n__g(activate(x0))), ACTIVATE(n__g(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: G_1, ACTIVATE_1, SEL_2 Compound Symbols: c3_1, c7_1, c6_2, c8_2, c6_1, c8_1 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, g(z0)), ACTIVATE(n__g(z0))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) S tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: G_1, ACTIVATE_1, SEL_2 Compound Symbols: c3_1, c7_1, c6_2, c8_2, c6_1, c8_1 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, g(z0)), ACTIVATE(n__g(z0))) by SEL(s(x0), cons(x1, n__g(0))) -> c6(SEL(x0, s(0)), ACTIVATE(n__g(0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(SEL(x0, s(s(g(z0)))), ACTIVATE(n__g(s(z0)))) SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, n__g(z0)), ACTIVATE(n__g(z0))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(0))) -> c6(SEL(x0, s(0)), ACTIVATE(n__g(0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(SEL(x0, s(s(g(z0)))), ACTIVATE(n__g(s(z0)))) SEL(s(x0), cons(x1, n__g(z0))) -> c6(SEL(x0, n__g(z0)), ACTIVATE(n__g(z0))) S tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: G_1, ACTIVATE_1, SEL_2 Compound Symbols: c3_1, c7_1, c6_2, c8_2, c6_1, c8_1 ---------------------------------------- (109) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(0))) -> c6(ACTIVATE(n__g(0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) S tuples: G(s(z0)) -> c3(G(z0)) ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: G_1, ACTIVATE_1, SEL_2 Compound Symbols: c3_1, c7_1, c6_2, c8_2, c6_1, c8_1 ---------------------------------------- (111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace G(s(z0)) -> c3(G(z0)) by G(s(s(y0))) -> c3(G(s(y0))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(0))) -> c6(ACTIVATE(n__g(0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) S tuples: ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: ACTIVATE_1, SEL_2, G_1 Compound Symbols: c7_1, c6_2, c8_2, c6_1, c8_1, c3_1 ---------------------------------------- (113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__f(z0)) -> c7(ACTIVATE(z0)) by ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(0))) -> c6(ACTIVATE(n__g(0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) S tuples: ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c6_1, c8_1, c3_1, c7_1 ---------------------------------------- (115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) by SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)), ACTIVATE(cons(y1, y2))) SEL(s(z0), cons(z1, n__g(y0))) -> c6(SEL(z0, n__g(y0)), ACTIVATE(n__g(y0))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2))), ACTIVATE(cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2)))), ACTIVATE(cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2)))), ACTIVATE(cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2)))), ACTIVATE(cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2)))), ACTIVATE(cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2))), ACTIVATE(cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(SEL(z0, n__g(n__f(n__f(y0)))), ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(SEL(z0, n__g(n__f(n__g(y0)))), ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(SEL(z0, n__g(n__f(y0))), ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(SEL(z0, n__g(n__g(n__f(y0)))), ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(SEL(z0, n__g(n__g(n__g(y0)))), ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(SEL(z0, n__g(n__g(y0))), ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0))), ACTIVATE(cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2)))), ACTIVATE(cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, n__f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, n__f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, n__f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, n__f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, n__f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, n__f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, n__f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, n__f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(0))) -> c6(ACTIVATE(n__g(0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)), ACTIVATE(cons(y1, y2))) SEL(s(z0), cons(z1, n__g(y0))) -> c6(SEL(z0, n__g(y0)), ACTIVATE(n__g(y0))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2))), ACTIVATE(cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2)))), ACTIVATE(cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2)))), ACTIVATE(cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2)))), ACTIVATE(cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2)))), ACTIVATE(cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2))), ACTIVATE(cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(SEL(z0, n__g(n__f(n__f(y0)))), ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(SEL(z0, n__g(n__f(n__g(y0)))), ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(SEL(z0, n__g(n__f(y0))), ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(SEL(z0, n__g(n__g(n__f(y0)))), ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(SEL(z0, n__g(n__g(n__g(y0)))), ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(SEL(z0, n__g(n__g(y0))), ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0))), ACTIVATE(cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2)))), ACTIVATE(cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, n__f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, n__f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, n__f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, n__f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, n__f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, n__f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, n__f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, n__f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) S tuples: ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: ACTIVATE_1, SEL_2, G_1 Compound Symbols: c8_2, c6_2, c6_1, c8_1, c3_1, c7_1 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 24 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(0))) -> c6(ACTIVATE(n__g(0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) S tuples: ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: ACTIVATE_1, SEL_2, G_1 Compound Symbols: c8_2, c6_2, c6_1, c8_1, c3_1, c7_1 ---------------------------------------- (119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__g(z0)) -> c8(G(z0), ACTIVATE(z0)) by ACTIVATE(n__g(n__g(y0))) -> c8(G(n__g(y0)), ACTIVATE(n__g(y0))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(G(n__g(n__f(n__f(y0)))), ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(G(n__g(n__f(n__g(y0)))), ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(G(n__g(n__f(y0))), ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(G(n__g(n__g(n__f(y0)))), ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(G(n__g(n__g(n__g(y0)))), ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(G(n__g(n__g(y0))), ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0))), ACTIVATE(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(n__f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(n__f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(n__f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(n__f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(n__f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(n__f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(n__f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(n__f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(0))) -> c6(ACTIVATE(n__g(0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(y0))) -> c8(G(n__g(y0)), ACTIVATE(n__g(y0))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(G(n__g(n__f(n__f(y0)))), ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(G(n__g(n__f(n__g(y0)))), ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(G(n__g(n__f(y0))), ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(G(n__g(n__g(n__f(y0)))), ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(G(n__g(n__g(n__g(y0)))), ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(G(n__g(n__g(y0))), ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0))), ACTIVATE(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(n__f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(n__f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(n__f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(n__f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(n__f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(n__f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(n__f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(n__f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(y0))) -> c8(G(n__g(y0)), ACTIVATE(n__g(y0))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(G(n__g(n__f(n__f(y0)))), ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(G(n__g(n__f(n__g(y0)))), ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(G(n__g(n__f(y0))), ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(G(n__g(n__g(n__f(y0)))), ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(G(n__g(n__g(n__g(y0)))), ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(G(n__g(n__g(y0))), ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0))), ACTIVATE(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(n__f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(n__f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(n__f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(n__f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(n__f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(n__f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(n__f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(n__f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c3_1, c7_1 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SEL(s(x0), cons(x1, n__g(0))) -> c6(ACTIVATE(n__g(0))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(y0))) -> c8(G(n__g(y0)), ACTIVATE(n__g(y0))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(G(n__g(n__f(n__f(y0)))), ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(G(n__g(n__f(n__g(y0)))), ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(G(n__g(n__f(y0))), ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(G(n__g(n__g(n__f(y0)))), ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(G(n__g(n__g(n__g(y0)))), ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(G(n__g(n__g(y0))), ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0))), ACTIVATE(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(n__f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(n__f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(n__f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(n__f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(n__f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(n__f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(n__f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(n__f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(y0))) -> c8(G(n__g(y0)), ACTIVATE(n__g(y0))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(G(n__g(n__f(n__f(y0)))), ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(G(n__g(n__f(n__g(y0)))), ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(G(n__g(n__f(y0))), ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(G(n__g(n__g(n__f(y0)))), ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(G(n__g(n__g(n__g(y0)))), ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(G(n__g(n__g(y0))), ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0))), ACTIVATE(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(n__f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(n__f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(n__f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(n__f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(n__f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(n__f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(n__f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(n__f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) K tuples:none Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c3_1, c7_1 ---------------------------------------- (123) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 16 trailing tuple parts ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) K tuples:none Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c3_1, c7_1 ---------------------------------------- (125) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) We considered the (Usable) Rules:none And the Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ACTIVATE(x_1)) = [1] POL(G(x_1)) = 0 POL(SEL(x_1, x_2)) = x_1 POL(activate(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = [1] + x_1 POL(f(x_1)) = 0 POL(g(x_1)) = 0 POL(n__f(x_1)) = 0 POL(n__g(x_1)) = 0 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c3_1, c7_1 ---------------------------------------- (127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(x0), cons(x1, n__f(z0))) -> c6(SEL(x0, f(z0)), ACTIVATE(n__f(z0))) by SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c3_1, c7_1 ---------------------------------------- (129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(x0), cons(x1, n__f(x2))) -> c6(ACTIVATE(n__f(x2))) by SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c3_1, c7_1 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) by SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c3_1, c7_1 ---------------------------------------- (133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__g(n__f(z0))) -> c8(G(f(z0)), ACTIVATE(n__f(z0))) by ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c3_1, c7_1 ---------------------------------------- (135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) by ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c3_1, c7_1 ---------------------------------------- (137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__g(n__f(x0))) -> c8(ACTIVATE(n__f(x0))) by ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c3_1, c7_1 ---------------------------------------- (139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) by ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c3_1, c7_1 ---------------------------------------- (141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(x0), cons(x1, n__g(s(z0)))) -> c6(ACTIVATE(n__g(s(z0)))) by SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c3_1, c7_1 ---------------------------------------- (143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(x0), cons(x1, n__g(z0))) -> c6(ACTIVATE(n__g(z0))) by SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__g(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__g(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) G(s(s(y0))) -> c3(G(s(y0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c3_1, c7_1 ---------------------------------------- (145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace G(s(s(y0))) -> c3(G(s(y0))) by G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__g(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c7_1, c3_1 ---------------------------------------- (147) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__f(n__f(y0))) -> c7(ACTIVATE(n__f(y0))) by ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__g(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c7_1, c3_1 ---------------------------------------- (149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__f(n__g(y0))) -> c7(ACTIVATE(n__g(y0))) by ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), 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c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__g(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c7_1, c3_1 ---------------------------------------- (151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__f(n__g(n__f(y0)))) -> c7(ACTIVATE(n__g(n__f(y0)))) by ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__g(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c7_1, c3_1 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) by SEL(s(s(z0)), cons(z1, cons(z2, n__f(y2)))) -> c6(SEL(s(z0), cons(z2, n__f(y2)))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__f(y2))))) -> c6(SEL(s(z0), cons(z2, n__f(n__f(y2))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(y2))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(y2))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(y2))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(y2))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(y2))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(y2))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(y2)))) -> c6(SEL(s(z0), cons(z2, n__g(y2)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__f(y3))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__f(y3))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__f(n__f(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__f(n__f(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__f(n__g(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__f(n__g(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(n__f(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(n__f(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(n__g(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(n__g(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(y3))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__f(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__f(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(y2)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(0))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(0))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(s(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(s(y3)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__f(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__f(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__f(y2)))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(y2)))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(y2)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__f(y2)))) -> c6(SEL(s(s(y0)), cons(z2, n__f(y2)))) SEL(s(s(s(s(y0)))), cons(z1, cons(z2, n__f(y2)))) -> c6(SEL(s(s(s(y0))), cons(z2, n__f(y2)))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(s(s(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(s(s(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(s(s(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(s(s(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) SEL(s(x0), cons(x1, n__g(x2))) -> c6(ACTIVATE(n__g(x2))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), 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cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__g(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, 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cons(z1, cons(z2, n__g(n__g(n__g(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(y2)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(0))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(0))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(s(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(s(y3)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__f(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__f(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__f(y2)))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(y2)))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(y2)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__f(y2)))) -> c6(SEL(s(s(y0)), cons(z2, n__f(y2)))) SEL(s(s(s(s(y0)))), cons(z1, cons(z2, n__f(y2)))) -> c6(SEL(s(s(s(y0))), cons(z2, n__f(y2)))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(s(s(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(s(s(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, 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tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c6_1, c8_2, c8_1, c7_1, c3_1 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__g(y0))) -> c6(ACTIVATE(n__g(y0))) by SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__g(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__g(n__g(x0))) -> c8(ACTIVATE(n__g(x0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__g(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, 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c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__f(y3))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__f(y3))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__f(n__f(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__f(n__f(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__f(n__g(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__f(n__g(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(n__f(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(n__f(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(n__g(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(n__g(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, 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-> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(y2)))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(y2)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__f(y2)))) -> c6(SEL(s(s(y0)), cons(z2, n__f(y2)))) SEL(s(s(s(s(y0)))), cons(z1, cons(z2, n__f(y2)))) -> c6(SEL(s(s(s(y0))), cons(z2, n__f(y2)))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(s(s(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(s(s(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(s(s(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(s(s(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c8_1, c7_1, c6_1, c3_1 ---------------------------------------- (157) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SEL(s(s(y0)), cons(z1, cons(y1, n__g(0)))) -> c6(SEL(s(y0), cons(y1, n__g(0)))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(y0))) -> c8(ACTIVATE(n__g(y0))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, 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cons(z2, n__g(n__f(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(s(s(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(s(s(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) by SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(y2))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(y2))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(y2))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(y2))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__f(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__f(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(s(s(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(s(s(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(s(s(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(s(s(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(y0))) -> c8(ACTIVATE(n__g(y0))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> 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c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (161) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) by SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(y0))) -> c8(ACTIVATE(n__g(y0))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, 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cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(s(s(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(s(s(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (163) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(s(y0)), cons(z1, cons(y1, n__g(s(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(s(y2))))) by SEL(s(s(z0)), cons(z1, cons(z2, n__g(s(s(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(s(s(y2)))))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, 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n__f(n__g(n__f(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__f(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__f(y2)))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(y2)))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(y2)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__f(y2)))) -> c6(SEL(s(s(y0)), cons(z2, n__f(y2)))) SEL(s(s(s(s(y0)))), cons(z1, cons(z2, n__f(y2)))) -> c6(SEL(s(s(s(y0))), cons(z2, n__f(y2)))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(s(s(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(s(s(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(s(s(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(s(s(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) by SEL(s(z0), cons(z1, n__f(n__f(n__f(y0))))) -> c6(ACTIVATE(n__f(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(y0))))) -> c6(ACTIVATE(n__f(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(y0))) -> c8(ACTIVATE(n__g(y0))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__g(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) 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SEL(s(s(s(s(y0)))), cons(z1, cons(z2, n__f(y2)))) -> c6(SEL(s(s(s(y0))), cons(z2, n__f(y2)))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(s(s(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(s(s(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(y2))))))) -> c6(SEL(s(z0), 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c6(SEL(s(z0), cons(z2, n__g(n__g(s(s(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) SEL(s(z0), cons(z1, n__f(n__f(n__f(y0))))) -> c6(ACTIVATE(n__f(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(y0))))) -> c6(ACTIVATE(n__f(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) by SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__f(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(s(s(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(y0))) -> c8(ACTIVATE(n__g(y0))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> 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cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(s(s(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(s(s(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) SEL(s(z0), cons(z1, n__f(n__f(n__f(y0))))) -> c6(ACTIVATE(n__f(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(y0))))) -> c6(ACTIVATE(n__f(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__f(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(s(s(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) by SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(y0))) -> c8(ACTIVATE(n__g(y0))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> 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c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> 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c6(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__f(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(s(s(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__g(n__g(y0))) -> c8(ACTIVATE(n__g(y0))) by ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__g(s(s(y0)))))) -> c8(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, 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c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__g(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__f(y3))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__f(y3))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__f(n__f(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__f(n__f(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__f(n__g(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__f(n__g(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(n__f(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(n__f(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(n__g(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(n__g(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(y3))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__f(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__f(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(y2)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(0))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(0))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(s(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(s(y3)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__f(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__f(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__f(y2)))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(y2)))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(y2)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__f(y2)))) -> c6(SEL(s(s(y0)), cons(z2, n__f(y2)))) SEL(s(s(s(s(y0)))), cons(z1, cons(z2, n__f(y2)))) -> c6(SEL(s(s(s(y0))), cons(z2, n__f(y2)))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(s(s(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(s(s(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(s(s(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(s(s(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) SEL(s(z0), cons(z1, n__f(n__f(n__f(y0))))) -> c6(ACTIVATE(n__f(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(y0))))) -> c6(ACTIVATE(n__f(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__f(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(s(s(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__g(s(s(y0)))))) -> c8(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (173) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__g(n__g(n__f(y0)))) -> c8(ACTIVATE(n__g(n__f(y0)))) by ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) 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c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(s(s(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(s(s(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(s(s(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(s(s(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, 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n__f(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__f(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(s(s(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__g(s(s(y0)))))) -> c8(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) K tuples: ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (175) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__g(s(s(y0)))) -> c8(G(s(s(y0)))) by ACTIVATE(n__g(s(s(s(y0))))) -> c8(G(s(s(s(y0))))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> 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cons(z2, n__g(n__g(n__f(n__g(n__g(n__g(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__g(y2)))))))) SEL(s(z0), cons(z1, n__f(n__f(n__f(y0))))) -> c6(ACTIVATE(n__f(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(y0))))) -> c6(ACTIVATE(n__f(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__f(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(s(s(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__g(s(s(y0)))))) -> c8(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(s(s(s(y0))))) -> c8(G(s(s(s(y0))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) K tuples: ACTIVATE(n__g(s(s(s(y0))))) -> c8(G(s(s(s(y0))))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (177) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) by ACTIVATE(n__g(n__f(n__f(n__f(y0))))) -> c8(ACTIVATE(n__f(n__f(n__f(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(y0))))) -> c8(ACTIVATE(n__f(n__f(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__g(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__f(y3))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__f(y3))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__f(n__f(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__f(n__f(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__f(n__g(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__f(n__g(y3)))))) SEL(s(s(s(y0))), cons(z1, 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c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__f(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__f(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(n__g(n__g(y2)))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(s(s(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(s(s(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__g(n__f(n__f(y2))))))))) SEL(s(s(z0)), cons(z1, cons(z2, 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c6(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__f(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(s(s(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__g(s(s(y0)))))) -> c8(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(s(s(s(y0))))) -> c8(G(s(s(s(y0))))) ACTIVATE(n__g(n__f(n__f(n__f(y0))))) -> c8(ACTIVATE(n__f(n__f(n__f(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(y0))))) -> c8(ACTIVATE(n__f(n__f(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__f(y0))))) -> c8(ACTIVATE(n__f(n__f(n__f(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(y0))))) -> c8(ACTIVATE(n__f(n__f(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) K tuples: ACTIVATE(n__g(s(s(s(y0))))) -> c8(G(s(s(s(y0))))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (179) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) by ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(s(s(y0)))))) -> c8(ACTIVATE(n__f(n__g(s(s(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(s(s(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, 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n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, 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c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__g(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) 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c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(s(s(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__f(y0))))) -> c8(ACTIVATE(n__f(n__f(n__f(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(y0))))) -> c8(ACTIVATE(n__f(n__f(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(s(s(y0)))))) -> c8(ACTIVATE(n__f(n__g(s(s(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(s(s(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) K tuples: ACTIVATE(n__g(s(s(s(y0))))) -> c8(G(s(s(s(y0))))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (181) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) by ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), 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c8(G(s(s(s(y0))))) ACTIVATE(n__g(n__f(n__f(n__f(y0))))) -> c8(ACTIVATE(n__f(n__f(n__f(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(y0))))) -> c8(ACTIVATE(n__f(n__f(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(y0))))))) -> 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ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(s(s(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__f(y0))))) -> c8(ACTIVATE(n__f(n__f(n__f(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(y0))))) -> c8(ACTIVATE(n__f(n__f(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(s(s(y0)))))) -> c8(ACTIVATE(n__f(n__g(s(s(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(s(s(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) K tuples: ACTIVATE(n__g(s(s(s(y0))))) -> c8(G(s(s(s(y0))))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (183) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(SEL(z0, f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) by SEL(s(s(y0)), cons(z1, n__f(n__f(z2)))) -> c6(SEL(s(y0), f(n__f(z2))), ACTIVATE(n__f(n__f(z2)))) SEL(s(s(s(y0))), cons(z1, n__f(n__f(z2)))) -> c6(SEL(s(s(y0)), f(n__f(z2))), ACTIVATE(n__f(n__f(z2)))) SEL(s(s(s(s(y0)))), cons(z1, n__f(n__f(z2)))) -> c6(SEL(s(s(s(y0))), f(n__f(z2))), ACTIVATE(n__f(n__f(z2)))) SEL(s(z0), cons(z1, n__f(n__f(n__f(y0))))) -> c6(SEL(z0, f(n__f(n__f(y0)))), ACTIVATE(n__f(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(y0))))) -> c6(SEL(z0, f(n__f(n__g(y0)))), ACTIVATE(n__f(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__f(y0))))))) -> c6(SEL(z0, f(n__f(n__g(n__f(n__f(y0)))))), ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__g(y0))))))) -> c6(SEL(z0, f(n__f(n__g(n__f(n__g(y0)))))), ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__f(n__g(n__f(y0))))), ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__f(y0))))))) -> c6(SEL(z0, f(n__f(n__g(n__g(n__f(y0)))))), ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__g(y0))))))) -> c6(SEL(z0, f(n__f(n__g(n__g(n__g(y0)))))), ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__f(n__g(n__g(y0))))), ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) SEL(s(s(s(s(s(y0))))), cons(z1, n__f(n__f(z2)))) -> c6(SEL(s(s(s(s(y0)))), f(n__f(z2))), ACTIVATE(n__f(n__f(z2)))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__g(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__f(y3))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__f(y3))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__f(n__f(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__f(n__f(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__f(n__g(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__f(n__g(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(n__f(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(n__f(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(n__g(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(n__g(y3)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(y3))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__f(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__f(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__f(n__g(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__f(n__g(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__g(y2)))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__g(y2)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(0))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(0))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__g(s(y3)))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__g(s(y3)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__f(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__f(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__f(y2)))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__f(y2)))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(n__g(y2))))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(n__g(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, n__f(n__g(n__g(y2)))))) -> c6(SEL(s(z0), cons(z2, n__f(n__g(n__g(y2)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__f(y2)))) -> c6(SEL(s(s(y0)), cons(z2, n__f(y2)))) SEL(s(s(s(s(y0)))), cons(z1, cons(z2, n__f(y2)))) -> c6(SEL(s(s(s(y0))), cons(z2, n__f(y2)))) SEL(s(s(z0)), cons(z1, cons(z2, n__g(n__g(n__f(n__f(y2))))))) -> c6(SEL(s(z0), cons(z2, n__g(n__g(n__f(n__f(y2))))))) SEL(s(s(z0)), cons(z1, cons(z2, 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c6(SEL(z0, f(n__f(n__g(n__f(n__f(y0)))))), ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__g(y0))))))) -> c6(SEL(z0, f(n__f(n__g(n__f(n__g(y0)))))), ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__f(n__g(n__f(y0))))), ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__f(y0))))))) -> c6(SEL(z0, f(n__f(n__g(n__g(n__f(y0)))))), ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__g(y0))))))) -> c6(SEL(z0, f(n__f(n__g(n__g(n__g(y0)))))), ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__f(n__g(n__g(y0))))), ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) SEL(s(s(s(s(s(y0))))), cons(z1, n__f(n__f(z2)))) -> c6(SEL(s(s(s(s(y0)))), f(n__f(z2))), ACTIVATE(n__f(n__f(z2)))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__f(y0))))) -> c8(ACTIVATE(n__f(n__f(n__f(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(y0))))) -> c8(ACTIVATE(n__f(n__f(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(s(s(y0)))))) -> c8(ACTIVATE(n__f(n__g(s(s(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(s(s(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) K tuples: ACTIVATE(n__g(s(s(s(y0))))) -> c8(G(s(s(s(y0))))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (185) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(SEL(z0, f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) by SEL(s(s(y0)), cons(z1, n__f(n__g(z2)))) -> c6(SEL(s(y0), f(n__g(z2))), ACTIVATE(n__f(n__g(z2)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(n__g(z2)))) -> c6(SEL(s(s(y0)), f(n__g(z2))), ACTIVATE(n__f(n__g(z2)))) SEL(s(s(s(s(y0)))), cons(z1, n__f(n__g(z2)))) -> c6(SEL(s(s(s(y0))), f(n__g(z2))), ACTIVATE(n__f(n__g(z2)))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__f(y0))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__f(y0)))))), ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(y0))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(y0)))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(y0))))))) -> c6(SEL(z0, f(n__g(n__g(n__g(n__f(y0)))))), ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(y0))))))) -> c6(SEL(z0, f(n__g(n__g(n__g(n__g(y0)))))), ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(s(s(y0)))))) -> c6(SEL(z0, f(n__g(s(s(y0))))), ACTIVATE(n__f(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(SEL(z0, f(n__g(n__f(n__g(n__f(n__f(y0))))))), ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(SEL(z0, f(n__g(n__f(n__g(n__f(n__g(y0))))))), ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(y0))))))) -> c6(SEL(z0, f(n__g(n__f(n__g(n__f(y0)))))), ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(SEL(z0, f(n__g(n__f(n__g(n__g(n__f(y0))))))), ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, 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c6(SEL(z0, f(n__g(n__g(s(s(y0)))))), ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(n__f(y0))))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(n__g(y0))))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) SEL(s(s(s(s(s(y0))))), cons(z1, n__f(n__g(z2)))) -> c6(SEL(s(s(s(s(y0)))), f(n__g(z2))), ACTIVATE(n__f(n__g(z2)))) SEL(s(s(s(s(s(s(y0)))))), cons(z1, n__f(n__g(z2)))) -> c6(SEL(s(s(s(s(s(y0))))), f(n__g(z2))), ACTIVATE(n__f(n__g(z2)))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, 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c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) 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c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__f(y0))))) 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ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(s(s(y0)))))) -> c8(ACTIVATE(n__f(n__g(s(s(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(s(s(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) K tuples: ACTIVATE(n__g(s(s(s(y0))))) -> c8(G(s(s(s(y0))))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (187) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(SEL(z0, f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) by SEL(s(s(y0)), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(y0), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(s(s(y0))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(y0)), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(s(y0)))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(s(y0))), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(SEL(z0, f(n__g(n__f(n__g(n__f(n__f(y0))))))), ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(SEL(z0, f(n__g(n__f(n__g(n__f(n__g(y0))))))), ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(y0))))))) -> c6(SEL(z0, f(n__g(n__f(n__g(n__f(y0)))))), ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(SEL(z0, f(n__g(n__f(n__g(n__g(n__f(y0))))))), ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(SEL(z0, f(n__g(n__f(n__g(n__g(n__g(y0))))))), ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(y0))))))) -> c6(SEL(z0, f(n__g(n__f(n__g(n__g(y0)))))), ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) SEL(s(s(s(s(s(y0))))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(s(s(y0)))), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(s(s(s(y0)))))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(s(s(s(y0))))), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(s(s(s(s(y0))))))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(s(s(s(s(y0)))))), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__g(s(s(y0)))))) -> 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cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(y0), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(y0))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(y0)), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(s(y0)))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(s(y0))), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(s(s(y0))))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(s(s(y0)))), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(s(s(s(y0)))))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(s(s(s(y0))))), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(s(s(s(s(y0))))))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(s(s(s(s(y0)))))), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> 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ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__f(y0))))) -> c8(ACTIVATE(n__f(n__f(n__f(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(y0))))) -> c8(ACTIVATE(n__f(n__f(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(s(s(y0)))))) -> c8(ACTIVATE(n__f(n__g(s(s(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(s(s(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) K tuples: ACTIVATE(n__g(s(s(s(y0))))) -> c8(G(s(s(s(y0))))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (189) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__f(n__f(y0)))) -> c6(ACTIVATE(n__f(n__f(y0)))) by SEL(s(z0), cons(z1, n__f(n__f(n__f(y0))))) -> c6(ACTIVATE(n__f(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(y0))))) -> c6(ACTIVATE(n__f(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) 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n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(n__g(y0))))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) SEL(s(s(s(s(s(y0))))), cons(z1, n__f(n__g(z2)))) -> c6(SEL(s(s(s(s(y0)))), f(n__g(z2))), ACTIVATE(n__f(n__g(z2)))) SEL(s(s(s(s(s(s(y0)))))), cons(z1, n__f(n__g(z2)))) -> c6(SEL(s(s(s(s(s(y0))))), f(n__g(z2))), ACTIVATE(n__f(n__g(z2)))) SEL(s(s(y0)), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(y0), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(y0))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(y0)), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(s(y0)))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(s(y0))), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(s(s(y0))))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(s(s(y0)))), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(s(s(s(y0)))))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(s(s(s(y0))))), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(s(s(s(s(y0))))))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(s(s(s(s(y0)))))), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) S tuples: ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__f(y0))))) -> c8(ACTIVATE(n__f(n__f(n__f(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(y0))))) -> c8(ACTIVATE(n__f(n__f(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(s(s(y0)))))) -> c8(ACTIVATE(n__f(n__g(s(s(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(s(s(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) K tuples: ACTIVATE(n__g(s(s(s(y0))))) -> c8(G(s(s(s(y0))))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (191) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__f(n__g(y0)))) -> c6(ACTIVATE(n__f(n__g(y0)))) by SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(s(s(y0)))))) -> c6(ACTIVATE(n__f(n__g(s(s(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(s(s(y0))))))) -> c6(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__f(z0)) -> f(activate(z0)) activate(n__g(z0)) -> g(activate(z0)) activate(z0) -> z0 f(z0) -> cons(z0, n__f(n__g(z0))) f(z0) -> n__f(z0) g(0) -> s(0) g(s(z0)) -> s(s(g(z0))) g(z0) -> n__g(z0) Tuples: SEL(s(x0), cons(x1, n__f(x2))) -> c6(SEL(x0, cons(activate(x2), n__f(n__g(activate(x2))))), ACTIVATE(n__f(x2))) SEL(s(x0), cons(x1, n__f(n__f(z0)))) -> c6(SEL(x0, f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) SEL(s(x0), cons(x1, n__f(n__g(z0)))) -> c6(SEL(x0, f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) SEL(s(x0), cons(x1, n__g(n__f(z0)))) -> c6(SEL(x0, g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) SEL(s(x0), cons(x1, n__g(n__g(z0)))) -> c6(SEL(x0, g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), ACTIVATE(n__f(n__f(z0)))) ACTIVATE(n__g(n__f(n__g(z0)))) -> c8(G(f(g(activate(z0)))), ACTIVATE(n__f(n__g(z0)))) ACTIVATE(n__g(n__g(n__f(z0)))) -> c8(G(g(f(activate(z0)))), ACTIVATE(n__g(n__f(z0)))) ACTIVATE(n__g(n__g(n__g(z0)))) -> c8(G(g(g(activate(z0)))), ACTIVATE(n__g(n__g(z0)))) ACTIVATE(n__g(n__g(z0))) -> c8(G(g(z0)), ACTIVATE(n__g(z0))) ACTIVATE(n__f(n__g(n__f(n__f(y0))))) -> c7(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(y0))))) -> c7(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(y0))))) -> c7(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__f(n__g(n__g(n__g(y0))))) -> c7(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(y0)))) -> c7(ACTIVATE(n__g(n__g(y0)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(y2)))) -> c6(SEL(s(y0), cons(y1, n__f(y2)))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__f(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__f(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__f(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__f(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(n__g(y2))))) -> c6(SEL(s(y0), cons(y1, n__g(n__g(y2))))) SEL(s(s(y0)), cons(z1, cons(y1, n__g(y2)))) -> c6(SEL(s(y0), cons(y1, n__g(y2)))) SEL(s(z0), cons(z1, n__g(n__f(n__f(y0))))) -> c6(ACTIVATE(n__g(n__f(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(y0))))) -> c6(ACTIVATE(n__g(n__f(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__f(y0)))) -> c6(ACTIVATE(n__g(n__f(y0)))) SEL(s(z0), cons(z1, n__g(n__g(n__f(y0))))) -> c6(ACTIVATE(n__g(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(y0))))) -> c6(ACTIVATE(n__g(n__g(n__g(y0))))) SEL(s(z0), cons(z1, n__g(n__g(y0)))) -> c6(ACTIVATE(n__g(n__g(y0)))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(y0))))) -> c6(ACTIVATE(n__f(n__g(n__f(y0))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__f(n__f(y0))))) -> c8(ACTIVATE(n__g(n__f(n__f(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(y0))))) -> c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(y0)), cons(z1, n__f(z2))) -> c6(SEL(s(y0), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(y0)))))) -> c6(SEL(z0, f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(y0))))) -> c6(SEL(z0, f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) SEL(s(s(s(y0))), cons(z1, n__f(z2))) -> c6(SEL(s(s(y0)), f(z2)), ACTIVATE(n__f(z2))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__f(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__g(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) SEL(s(z0), cons(z1, n__g(s(s(y0))))) -> c6(ACTIVATE(n__g(s(s(y0))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__f(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__f(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(n__g(y0))))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__g(n__f(n__g(n__g(y0)))))) -> c6(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> 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c6(SEL(z0, f(n__g(n__f(n__g(n__g(n__g(y0))))))), ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__f(n__g(n__g(y0))))))) -> c6(SEL(z0, f(n__g(n__f(n__g(n__g(y0)))))), ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c6(SEL(z0, f(n__g(n__g(n__g(n__f(n__f(y0))))))), ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c6(SEL(z0, f(n__g(n__g(n__g(n__f(n__g(y0))))))), ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c6(SEL(z0, f(n__g(n__g(n__g(n__g(n__f(y0))))))), ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c6(SEL(z0, f(n__g(n__g(n__g(n__g(n__g(y0))))))), ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(s(s(y0))))))) -> c6(SEL(z0, f(n__g(n__g(s(s(y0)))))), ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(n__f(y0))))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) SEL(s(z0), cons(z1, n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c6(SEL(z0, f(n__g(n__g(n__f(n__g(n__g(y0))))))), ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) SEL(s(s(s(s(s(y0))))), cons(z1, n__f(n__g(z2)))) -> c6(SEL(s(s(s(s(y0)))), f(n__g(z2))), ACTIVATE(n__f(n__g(z2)))) SEL(s(s(s(s(s(s(y0)))))), cons(z1, n__f(n__g(z2)))) -> c6(SEL(s(s(s(s(s(y0))))), f(n__g(z2))), ACTIVATE(n__f(n__g(z2)))) SEL(s(s(y0)), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(y0), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(y0))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(y0)), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(s(y0)))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(s(y0))), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(s(s(y0))))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(s(s(y0)))), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(s(s(s(y0)))))), cons(z1, n__f(n__g(n__f(z2))))) -> c6(SEL(s(s(s(s(s(y0))))), f(n__g(n__f(z2)))), ACTIVATE(n__f(n__g(n__f(z2))))) SEL(s(s(s(s(s(s(s(y0))))))), cons(z1, 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c8(ACTIVATE(n__g(n__f(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(y0))))) -> c8(ACTIVATE(n__g(n__g(n__f(y0))))) ACTIVATE(n__g(n__g(n__g(n__g(y0))))) -> c8(ACTIVATE(n__g(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(y0)))) -> c8(ACTIVATE(n__g(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(y0)))) -> c8(G(f(n__f(y0))), ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__g(n__f(n__g(y0)))) -> c8(G(f(n__g(y0))), ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0)))))) -> c8(G(f(n__g(n__f(n__f(y0))))), ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0)))))) -> c8(G(f(n__g(n__f(n__g(y0))))), ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(y0))))) -> c8(G(f(n__g(n__f(y0)))), ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0)))))) -> c8(G(f(n__g(n__g(n__f(y0))))), ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0)))))) -> c8(G(f(n__g(n__g(n__g(y0))))), ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(y0))))) -> c8(G(f(n__g(n__g(y0)))), ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__g(n__g(s(s(y0))))) -> c8(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) G(s(s(s(y0)))) -> c3(G(s(s(y0)))) ACTIVATE(n__f(n__f(n__f(y0)))) -> c7(ACTIVATE(n__f(n__f(y0)))) ACTIVATE(n__f(n__f(n__g(y0)))) -> c7(ACTIVATE(n__f(n__g(y0)))) ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__f(y0))))) -> c7(ACTIVATE(n__f(n__g(n__f(y0))))) ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__f(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__f(n__g(n__g(y0))))) -> c7(ACTIVATE(n__f(n__g(n__g(y0))))) ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(s(s(y0))))) -> c7(ACTIVATE(n__g(s(s(y0))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__f(y0)))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0)))))) -> c7(ACTIVATE(n__g(n__f(n__g(n__g(y0)))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__f(n__g(n__g(s(s(y0)))))) -> c7(ACTIVATE(n__g(n__g(s(s(y0)))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0))))))) -> c7(ACTIVATE(n__g(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__f(y0))))) -> c8(ACTIVATE(n__f(n__f(n__f(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(y0))))) -> c8(ACTIVATE(n__f(n__f(n__g(y0))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__f(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__f(y0)))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__f(n__g(n__g(y0)))))) -> c8(ACTIVATE(n__f(n__f(n__g(n__g(y0)))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(s(s(y0)))))) -> c8(ACTIVATE(n__f(n__g(s(s(y0)))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__f(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__f(y0))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__f(n__g(n__g(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__f(n__g(n__g(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__f(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__g(n__g(n__g(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(s(s(y0))))))) -> c8(ACTIVATE(n__f(n__g(n__g(s(s(y0))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__f(y0)))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__f(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(n__g(y0))))))))) ACTIVATE(n__g(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) -> c8(ACTIVATE(n__f(n__g(n__g(n__f(n__g(n__g(y0)))))))) K tuples: ACTIVATE(n__g(s(s(s(y0))))) -> c8(G(s(s(s(y0))))) Defined Rule Symbols: activate_1, f_1, g_1 Defined Pair Symbols: SEL_2, ACTIVATE_1, G_1 Compound Symbols: c6_2, c8_2, c7_1, c6_1, c8_1, c3_1 ---------------------------------------- (193) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (194) 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) -> cons(X, n__f(n__g(X))) [1] g(0) -> s(0) [1] g(s(X)) -> s(s(g(X))) [1] sel(0, cons(X, Y)) -> X [1] sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) [1] f(X) -> n__f(X) [1] g(X) -> n__g(X) [1] activate(n__f(X)) -> f(activate(X)) [1] activate(n__g(X)) -> g(activate(X)) [1] activate(X) -> X [1] Rewrite Strategy: INNERMOST ---------------------------------------- (195) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (196) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(X) -> cons(X, n__f(n__g(X))) [1] g(0) -> s(0) [1] g(s(X)) -> s(s(g(X))) [1] sel(0, cons(X, Y)) -> X [1] sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) [1] f(X) -> n__f(X) [1] g(X) -> n__g(X) [1] activate(n__f(X)) -> f(activate(X)) [1] activate(n__g(X)) -> g(activate(X)) [1] activate(X) -> X [1] The TRS has the following type information: f :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s cons :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s n__f :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s n__g :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s g :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s 0 :: n__g:n__f:cons:0:s s :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s sel :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s activate :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s Rewrite Strategy: INNERMOST ---------------------------------------- (197) 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: sel_2 (c) The following functions are completely defined: activate_1 g_1 f_1 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (198) 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) -> cons(X, n__f(n__g(X))) [1] g(0) -> s(0) [1] g(s(X)) -> s(s(g(X))) [1] sel(0, cons(X, Y)) -> X [1] sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) [1] f(X) -> n__f(X) [1] g(X) -> n__g(X) [1] activate(n__f(X)) -> f(activate(X)) [1] activate(n__g(X)) -> g(activate(X)) [1] activate(X) -> X [1] The TRS has the following type information: f :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s cons :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s n__f :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s n__g :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s g :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s 0 :: n__g:n__f:cons:0:s s :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s sel :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s activate :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s Rewrite Strategy: INNERMOST ---------------------------------------- (199) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (200) 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) -> cons(X, n__f(n__g(X))) [1] g(0) -> s(0) [1] g(s(X)) -> s(s(g(X))) [1] sel(0, cons(X, Y)) -> X [1] sel(s(X), cons(Y, n__f(X'))) -> sel(X, f(activate(X'))) [2] sel(s(X), cons(Y, n__g(X''))) -> sel(X, g(activate(X''))) [2] sel(s(X), cons(Y, Z)) -> sel(X, Z) [2] f(X) -> n__f(X) [1] g(X) -> n__g(X) [1] activate(n__f(n__f(X1))) -> f(f(activate(X1))) [2] activate(n__f(n__g(X2))) -> f(g(activate(X2))) [2] activate(n__f(X)) -> f(X) [2] activate(n__g(n__f(X3))) -> g(f(activate(X3))) [2] activate(n__g(n__g(X4))) -> g(g(activate(X4))) [2] activate(n__g(X)) -> g(X) [2] activate(X) -> X [1] The TRS has the following type information: f :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s cons :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s n__f :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s n__g :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s g :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s 0 :: n__g:n__f:cons:0:s s :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s sel :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s activate :: n__g:n__f:cons:0:s -> n__g:n__f:cons:0:s Rewrite Strategy: INNERMOST ---------------------------------------- (201) 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 ---------------------------------------- (202) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 2 }-> g(X) :|: z = 1 + X, X >= 0 activate(z) -{ 2 }-> g(g(activate(X4))) :|: z = 1 + (1 + X4), X4 >= 0 activate(z) -{ 2 }-> g(f(activate(X3))) :|: z = 1 + (1 + X3), X3 >= 0 activate(z) -{ 2 }-> f(X) :|: z = 1 + X, X >= 0 activate(z) -{ 2 }-> f(g(activate(X2))) :|: z = 1 + (1 + X2), X2 >= 0 activate(z) -{ 2 }-> f(f(activate(X1))) :|: X1 >= 0, z = 1 + (1 + X1) f(z) -{ 1 }-> 1 + X :|: X >= 0, z = X f(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X g(z) -{ 1 }-> 1 + X :|: X >= 0, z = X g(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 1 }-> 1 + (1 + g(X)) :|: z = 1 + X, X >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 2 }-> sel(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z sel(z, z') -{ 2 }-> sel(X, g(activate(X''))) :|: z = 1 + X, Y >= 0, X >= 0, X'' >= 0, z' = 1 + Y + (1 + X'') sel(z, z') -{ 2 }-> sel(X, f(activate(X'))) :|: z = 1 + X, Y >= 0, z' = 1 + Y + (1 + X'), X >= 0, X' >= 0 ---------------------------------------- (203) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: f(z) -{ 1 }-> 1 + X :|: X >= 0, z = X f(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X ---------------------------------------- (204) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 2 }-> g(X) :|: z = 1 + X, X >= 0 activate(z) -{ 2 }-> g(g(activate(X4))) :|: z = 1 + (1 + X4), X4 >= 0 activate(z) -{ 2 }-> g(f(activate(X3))) :|: z = 1 + (1 + X3), X3 >= 0 activate(z) -{ 2 }-> f(g(activate(X2))) :|: z = 1 + (1 + X2), X2 >= 0 activate(z) -{ 2 }-> f(f(activate(X1))) :|: X1 >= 0, z = 1 + (1 + X1) activate(z) -{ 3 }-> 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X' activate(z) -{ 3 }-> 1 + X' + (1 + (1 + X')) :|: z = 1 + X, X >= 0, X' >= 0, X = X' f(z) -{ 1 }-> 1 + X :|: X >= 0, z = X f(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X g(z) -{ 1 }-> 1 + X :|: X >= 0, z = X g(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 1 }-> 1 + (1 + g(X)) :|: z = 1 + X, X >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 2 }-> sel(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z sel(z, z') -{ 2 }-> sel(X, g(activate(X''))) :|: z = 1 + X, Y >= 0, X >= 0, X'' >= 0, z' = 1 + Y + (1 + X'') sel(z, z') -{ 2 }-> sel(X, f(activate(X'))) :|: z = 1 + X, Y >= 0, z' = 1 + Y + (1 + X'), X >= 0, X' >= 0 ---------------------------------------- (205) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (206) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(z - 1) :|: z - 1 >= 0 activate(z) -{ 2 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 3 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + z :|: z >= 0 f(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 1 }-> 1 + z :|: z >= 0 g(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 1 }-> 1 + (1 + g(z - 1)) :|: z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sel(z, z') -{ 2 }-> sel(z - 1, g(activate(X''))) :|: Y >= 0, z - 1 >= 0, X'' >= 0, z' = 1 + Y + (1 + X'') sel(z, z') -{ 2 }-> sel(z - 1, f(activate(X'))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0 ---------------------------------------- (207) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { g } { f } { activate } { sel } ---------------------------------------- (208) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(z - 1) :|: z - 1 >= 0 activate(z) -{ 2 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 3 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + z :|: z >= 0 f(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 1 }-> 1 + z :|: z >= 0 g(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 1 }-> 1 + (1 + g(z - 1)) :|: z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sel(z, z') -{ 2 }-> sel(z - 1, g(activate(X''))) :|: Y >= 0, z - 1 >= 0, X'' >= 0, z' = 1 + Y + (1 + X'') sel(z, z') -{ 2 }-> sel(z - 1, f(activate(X'))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0 Function symbols to be analyzed: {g}, {f}, {activate}, {sel} ---------------------------------------- (209) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (210) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(z - 1) :|: z - 1 >= 0 activate(z) -{ 2 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 3 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + z :|: z >= 0 f(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 1 }-> 1 + z :|: z >= 0 g(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 1 }-> 1 + (1 + g(z - 1)) :|: z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sel(z, z') -{ 2 }-> sel(z - 1, g(activate(X''))) :|: Y >= 0, z - 1 >= 0, X'' >= 0, z' = 1 + Y + (1 + X'') sel(z, z') -{ 2 }-> sel(z - 1, f(activate(X'))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0 Function symbols to be analyzed: {g}, {f}, {activate}, {sel} ---------------------------------------- (211) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 2*z ---------------------------------------- (212) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(z - 1) :|: z - 1 >= 0 activate(z) -{ 2 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 3 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + z :|: z >= 0 f(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 1 }-> 1 + z :|: z >= 0 g(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 1 }-> 1 + (1 + g(z - 1)) :|: z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sel(z, z') -{ 2 }-> sel(z - 1, g(activate(X''))) :|: Y >= 0, z - 1 >= 0, X'' >= 0, z' = 1 + Y + (1 + X'') sel(z, z') -{ 2 }-> sel(z - 1, f(activate(X'))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0 Function symbols to be analyzed: {g}, {f}, {activate}, {sel} Previous analysis results are: g: runtime: ?, size: O(n^1) [1 + 2*z] ---------------------------------------- (213) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (214) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(z - 1) :|: z - 1 >= 0 activate(z) -{ 2 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 3 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + z :|: z >= 0 f(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 1 }-> 1 + z :|: z >= 0 g(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 1 }-> 1 + (1 + g(z - 1)) :|: z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sel(z, z') -{ 2 }-> sel(z - 1, g(activate(X''))) :|: Y >= 0, z - 1 >= 0, X'' >= 0, z' = 1 + Y + (1 + X'') sel(z, z') -{ 2 }-> sel(z - 1, f(activate(X'))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0 Function symbols to be analyzed: {f}, {activate}, {sel} Previous analysis results are: g: runtime: O(n^1) [1 + z], size: O(n^1) [1 + 2*z] ---------------------------------------- (215) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (216) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 3 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + z :|: z >= 0 f(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 1 }-> 1 + z :|: z >= 0 g(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sel(z, z') -{ 2 }-> sel(z - 1, g(activate(X''))) :|: Y >= 0, z - 1 >= 0, X'' >= 0, z' = 1 + Y + (1 + X'') sel(z, z') -{ 2 }-> sel(z - 1, f(activate(X'))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0 Function symbols to be analyzed: {f}, {activate}, {sel} Previous analysis results are: g: runtime: O(n^1) [1 + z], size: O(n^1) [1 + 2*z] ---------------------------------------- (217) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 2*z ---------------------------------------- (218) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 3 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + z :|: z >= 0 f(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 1 }-> 1 + z :|: z >= 0 g(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sel(z, z') -{ 2 }-> sel(z - 1, g(activate(X''))) :|: Y >= 0, z - 1 >= 0, X'' >= 0, z' = 1 + Y + (1 + X'') sel(z, z') -{ 2 }-> sel(z - 1, f(activate(X'))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0 Function symbols to be analyzed: {f}, {activate}, {sel} Previous analysis results are: g: runtime: O(n^1) [1 + z], size: O(n^1) [1 + 2*z] f: runtime: ?, size: O(n^1) [3 + 2*z] ---------------------------------------- (219) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (220) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 3 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + z :|: z >= 0 f(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 1 }-> 1 + z :|: z >= 0 g(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sel(z, z') -{ 2 }-> sel(z - 1, g(activate(X''))) :|: Y >= 0, z - 1 >= 0, X'' >= 0, z' = 1 + Y + (1 + X'') sel(z, z') -{ 2 }-> sel(z - 1, f(activate(X'))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0 Function symbols to be analyzed: {activate}, {sel} Previous analysis results are: g: runtime: O(n^1) [1 + z], size: O(n^1) [1 + 2*z] f: runtime: O(1) [1], size: O(n^1) [3 + 2*z] ---------------------------------------- (221) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (222) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 3 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + z :|: z >= 0 f(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 1 }-> 1 + z :|: z >= 0 g(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sel(z, z') -{ 2 }-> sel(z - 1, g(activate(X''))) :|: Y >= 0, z - 1 >= 0, X'' >= 0, z' = 1 + Y + (1 + X'') sel(z, z') -{ 2 }-> sel(z - 1, f(activate(X'))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0 Function symbols to be analyzed: {activate}, {sel} Previous analysis results are: g: runtime: O(n^1) [1 + z], size: O(n^1) [1 + 2*z] f: runtime: O(1) [1], size: O(n^1) [3 + 2*z] ---------------------------------------- (223) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: activate after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (224) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 3 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + z :|: z >= 0 f(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 1 }-> 1 + z :|: z >= 0 g(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sel(z, z') -{ 2 }-> sel(z - 1, g(activate(X''))) :|: Y >= 0, z - 1 >= 0, X'' >= 0, z' = 1 + Y + (1 + X'') sel(z, z') -{ 2 }-> sel(z - 1, f(activate(X'))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0 Function symbols to be analyzed: {activate}, {sel} Previous analysis results are: g: runtime: O(n^1) [1 + z], size: O(n^1) [1 + 2*z] f: runtime: O(1) [1], size: O(n^1) [3 + 2*z] activate: runtime: ?, size: EXP ---------------------------------------- (225) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: activate after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (226) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> g(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> g(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(g(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 2 }-> f(f(activate(z - 2))) :|: z - 2 >= 0 activate(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 3 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + z :|: z >= 0 f(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 g(z) -{ 1 }-> 1 + z :|: z >= 0 g(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1) + 1, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sel(z, z') -{ 2 }-> sel(z - 1, g(activate(X''))) :|: Y >= 0, z - 1 >= 0, X'' >= 0, z' = 1 + Y + (1 + X'') sel(z, z') -{ 2 }-> sel(z - 1, f(activate(X'))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0 Function symbols to be analyzed: {activate}, {sel} Previous analysis results are: g: runtime: O(n^1) [1 + z], size: O(n^1) [1 + 2*z] f: runtime: O(1) [1], size: O(n^1) [3 + 2*z] activate: runtime: INF, size: EXP ---------------------------------------- (227) 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: sel(v0, v1) -> null_sel [0] And the following fresh constants: null_sel ---------------------------------------- (228) 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) -> cons(X, n__f(n__g(X))) [1] g(0) -> s(0) [1] g(s(X)) -> s(s(g(X))) [1] sel(0, cons(X, Y)) -> X [1] sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) [1] f(X) -> n__f(X) [1] g(X) -> n__g(X) [1] activate(n__f(X)) -> f(activate(X)) [1] activate(n__g(X)) -> g(activate(X)) [1] activate(X) -> X [1] sel(v0, v1) -> null_sel [0] The TRS has the following type information: f :: n__g:n__f:cons:0:s:null_sel -> n__g:n__f:cons:0:s:null_sel cons :: n__g:n__f:cons:0:s:null_sel -> n__g:n__f:cons:0:s:null_sel -> n__g:n__f:cons:0:s:null_sel n__f :: n__g:n__f:cons:0:s:null_sel -> n__g:n__f:cons:0:s:null_sel n__g :: n__g:n__f:cons:0:s:null_sel -> n__g:n__f:cons:0:s:null_sel g :: n__g:n__f:cons:0:s:null_sel -> n__g:n__f:cons:0:s:null_sel 0 :: n__g:n__f:cons:0:s:null_sel s :: n__g:n__f:cons:0:s:null_sel -> n__g:n__f:cons:0:s:null_sel sel :: n__g:n__f:cons:0:s:null_sel -> n__g:n__f:cons:0:s:null_sel -> n__g:n__f:cons:0:s:null_sel activate :: n__g:n__f:cons:0:s:null_sel -> n__g:n__f:cons:0:s:null_sel null_sel :: n__g:n__f:cons:0:s:null_sel Rewrite Strategy: INNERMOST ---------------------------------------- (229) 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_sel => 0 ---------------------------------------- (230) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> g(activate(X)) :|: z = 1 + X, X >= 0 activate(z) -{ 1 }-> f(activate(X)) :|: z = 1 + X, X >= 0 f(z) -{ 1 }-> 1 + X :|: X >= 0, z = X f(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X g(z) -{ 1 }-> 1 + X :|: X >= 0, z = X g(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 1 }-> 1 + (1 + g(X)) :|: z = 1 + X, X >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 1 }-> sel(X, activate(Z)) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z sel(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity.