KILLED proof of input_IVaEzEa7KW.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) InliningProof [UPPER BOUND(ID), 87 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 245 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 68 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 176 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 2086 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 40 ms] (38) CpxRNTS (39) CompletionProof [UPPER BOUND(ID), 0 ms] (40) CpxTypedWeightedCompleteTrs (41) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (44) CdtProblem (45) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 120 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 6 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 58 ms] (92) CdtProblem (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 8 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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 12 ms] (118) CdtProblem (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 32 ms] (120) CdtProblem (121) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 12 ms] (122) CdtProblem (123) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 9 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), 17 ms] (132) CdtProblem (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 3 ms] (134) CdtProblem (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 6 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), 0 ms] (144) CdtProblem (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 16 ms] (146) CdtProblem (147) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 31 ms] (150) CdtProblem (151) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 216 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 71 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 59 ms] (158) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: 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) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: 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 ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) 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 ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: 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 ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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 ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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 ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 ---------------------------------------- (14) 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 ---------------------------------------- (15) 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 ---------------------------------------- (16) 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 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: 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 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { g } { f } { activate } { sel } ---------------------------------------- (20) 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} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: 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} ---------------------------------------- (23) 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 ---------------------------------------- (24) 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] ---------------------------------------- (25) 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 ---------------------------------------- (26) 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] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: 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] ---------------------------------------- (29) 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 ---------------------------------------- (30) 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] ---------------------------------------- (31) 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 ---------------------------------------- (32) 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] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: 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] ---------------------------------------- (35) 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: ? ---------------------------------------- (36) 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 ---------------------------------------- (37) 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: ? ---------------------------------------- (38) 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 ---------------------------------------- (39) 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 ---------------------------------------- (40) 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 ---------------------------------------- (41) 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 ---------------------------------------- (42) 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. ---------------------------------------- (43) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (44) 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 ---------------------------------------- (45) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: F(z0) -> c1 G(0) -> c2 G(z0) -> c4 ACTIVATE(z0) -> c9 SEL(0, cons(z0, z1)) -> c5 F(z0) -> c ---------------------------------------- (46) 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 ---------------------------------------- (47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (48) 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 ---------------------------------------- (49) 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)) ---------------------------------------- (50) 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 ---------------------------------------- (51) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) 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) = [2] POL(ACTIVATE(x_1)) = [1] POL(G(x_1)) = 0 POL(SEL(x_1, x_2)) = x_1^2 POL(activate(x_1)) = 0 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)) = x_1 POL(f(x_1)) = [1] POL(g(x_1)) = [1] POL(n__f(x_1)) = 0 POL(n__g(x_1)) = 0 POL(s(x_1)) = [2] + x_1 ---------------------------------------- (52) 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 ---------------------------------------- (53) 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)) ---------------------------------------- (54) 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 ---------------------------------------- (55) 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)) ---------------------------------------- (56) 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 ---------------------------------------- (57) 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))) ---------------------------------------- (58) 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 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (60) 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 ---------------------------------------- (61) 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))) ---------------------------------------- (62) 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 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (64) 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 ---------------------------------------- (65) 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))) ---------------------------------------- (66) 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 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (68) 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 ---------------------------------------- (69) 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))) ---------------------------------------- (70) 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 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (72) 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 ---------------------------------------- (73) 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))) ---------------------------------------- (74) 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 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (76) 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 ---------------------------------------- (77) 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))) ---------------------------------------- (78) 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 ---------------------------------------- (79) 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)))) ---------------------------------------- (80) 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 ---------------------------------------- (81) 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))))) ---------------------------------------- (82) 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 ---------------------------------------- (83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 24 trailing tuple parts ---------------------------------------- (84) 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 ---------------------------------------- (85) 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))))) ---------------------------------------- (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: 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 ---------------------------------------- (87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SEL(s(x0), cons(x1, n__g(0))) -> c6(ACTIVATE(n__g(0))) ---------------------------------------- (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: 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 ---------------------------------------- (89) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 16 trailing tuple parts ---------------------------------------- (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: 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 ---------------------------------------- (91) 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 ---------------------------------------- (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: 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 ---------------------------------------- (93) 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))) ---------------------------------------- (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: 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 ---------------------------------------- (95) 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))))) ---------------------------------------- (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: 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 ---------------------------------------- (97) 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)))))) ---------------------------------------- (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: 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 ---------------------------------------- (99) 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))))) ---------------------------------------- (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: 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 ---------------------------------------- (101) 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))))) ---------------------------------------- (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: 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 ---------------------------------------- (103) 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))))) ---------------------------------------- (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: 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 ---------------------------------------- (105) 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)))))) ---------------------------------------- (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: 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 ---------------------------------------- (107) 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))))) ---------------------------------------- (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: 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 ---------------------------------------- (109) 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))))))) ---------------------------------------- (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: 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 ---------------------------------------- (111) 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)))) ---------------------------------------- (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: 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 ---------------------------------------- (113) 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))))) ---------------------------------------- (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, 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 ---------------------------------------- (115) 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))))))) ---------------------------------------- (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: 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), 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__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 ---------------------------------------- (117) 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)))))) ---------------------------------------- (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: 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 ---------------------------------------- (119) 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)))))))) ---------------------------------------- (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__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)), 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)))))) 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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 ---------------------------------------- (121) 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))))))) ---------------------------------------- (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__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))))) 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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, 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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__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, 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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 ---------------------------------------- (123) 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)))) ---------------------------------------- (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__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)))))) -> 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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 ---------------------------------------- (125) 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)))))))) ---------------------------------------- (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__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))))) -> 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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 ---------------------------------------- (127) 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)))))) ---------------------------------------- (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__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)))))) -> 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))))))) -> 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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 ---------------------------------------- (129) 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)))))) ---------------------------------------- (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)))) 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)))))) -> 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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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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 ---------------------------------------- (131) 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)))))) ---------------------------------------- (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)))) 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, 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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 ---------------------------------------- (133) 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)))))))) ---------------------------------------- (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, 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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, 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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, 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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 ---------------------------------------- (135) 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))))))) ---------------------------------------- (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)))) 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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)))))))) 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 ---------------------------------------- (137) 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))))))) ---------------------------------------- (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)))) 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__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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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 ---------------------------------------- (139) 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)))))) ---------------------------------------- (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)))) 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))))) 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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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 ---------------------------------------- (141) 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))))) ---------------------------------------- (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)))) 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, 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)))))) 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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), 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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 ---------------------------------------- (143) 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)))))) ---------------------------------------- (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)))) 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, 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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 ---------------------------------------- (145) 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)))))))) ---------------------------------------- (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)))) 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)))))) -> 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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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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)))))) 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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 ---------------------------------------- (147) 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))))))) ---------------------------------------- (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)))) ACTIVATE(n__g(n__f(n__f(z0)))) -> c8(G(f(f(activate(z0)))), 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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 ---------------------------------------- (149) 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)))) ---------------------------------------- (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)))) 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)))))) 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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))))) 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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 ---------------------------------------- (151) 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, 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(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)))) ---------------------------------------- (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)))) 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)))))) 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)))))) 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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 ---------------------------------------- (153) 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))))) ---------------------------------------- (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)))) 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)))))) -> 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)))))))) 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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))))))) -> 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 ---------------------------------------- (155) 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)))))) ---------------------------------------- (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__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))))), 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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 ---------------------------------------- (157) 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)))))))) ---------------------------------------- (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(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)))))) -> 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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