/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- KILLED proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 54 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 3 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 449 ms] (12) BOUNDS(1, INF) (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (14) TRS for Loop Detection (15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxTRS (17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (18) CpxRelTRS (19) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxWeightedTrs (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxTypedWeightedTrs (23) CompletionProof [UPPER BOUND(ID), 0 ms] (24) CpxTypedWeightedCompleteTrs (25) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (38) CdtProblem (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (44) CdtProblem (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (64) CdtProblem (65) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 2 ms] (126) CdtProblem (127) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(s(s(x1)))) -> s(s(a(a(x1)))) b(b(a(a(b(b(s(s(x1)))))))) -> a(a(b(b(s(s(a(a(x1)))))))) b(b(a(a(b(b(b(b(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) a(a(b(b(a(a(a(a(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(s(s(x1)))) -> s(s(a(a(x1)))) b(b(a(a(b(b(s(s(x1)))))))) -> a(a(b(b(s(s(a(a(x1)))))))) b(b(a(a(b(b(b(b(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) a(a(b(b(a(a(a(a(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(s(s(x1)))) -> s(s(a(a(x1)))) b(b(a(a(b(b(s(s(x1)))))))) -> a(a(b(b(s(s(a(a(x1)))))))) b(b(a(a(b(b(b(b(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) a(a(b(b(a(a(a(a(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(s(s(x1)))) -> s(s(a(a(x1)))) b(b(a(a(b(b(s(s(x1)))))))) -> a(a(b(b(s(s(a(a(x1)))))))) b(b(a(a(b(b(b(b(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) a(a(b(b(a(a(a(a(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: a(a(s(s(x1)))) -> s(s(a(a(x1)))) b(b(a(a(b(b(s(s(x1)))))))) -> a(a(b(b(s(s(a(a(x1)))))))) b(b(a(a(b(b(b(b(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) a(a(b(b(a(a(a(a(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) Types: a :: s:cons_a:cons_b -> s:cons_a:cons_b s :: s:cons_a:cons_b -> s:cons_a:cons_b b :: s:cons_a:cons_b -> s:cons_a:cons_b encArg :: s:cons_a:cons_b -> s:cons_a:cons_b cons_a :: s:cons_a:cons_b -> s:cons_a:cons_b cons_b :: s:cons_a:cons_b -> s:cons_a:cons_b encode_a :: s:cons_a:cons_b -> s:cons_a:cons_b encode_s :: s:cons_a:cons_b -> s:cons_a:cons_b encode_b :: s:cons_a:cons_b -> s:cons_a:cons_b hole_s:cons_a:cons_b1_0 :: s:cons_a:cons_b gen_s:cons_a:cons_b2_0 :: Nat -> s:cons_a:cons_b ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: a, b, encArg They will be analysed ascendingly in the following order: a = b a < encArg b < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: a(a(s(s(x1)))) -> s(s(a(a(x1)))) b(b(a(a(b(b(s(s(x1)))))))) -> a(a(b(b(s(s(a(a(x1)))))))) b(b(a(a(b(b(b(b(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) a(a(b(b(a(a(a(a(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) Types: a :: s:cons_a:cons_b -> s:cons_a:cons_b s :: s:cons_a:cons_b -> s:cons_a:cons_b b :: s:cons_a:cons_b -> s:cons_a:cons_b encArg :: s:cons_a:cons_b -> s:cons_a:cons_b cons_a :: s:cons_a:cons_b -> s:cons_a:cons_b cons_b :: s:cons_a:cons_b -> s:cons_a:cons_b encode_a :: s:cons_a:cons_b -> s:cons_a:cons_b encode_s :: s:cons_a:cons_b -> s:cons_a:cons_b encode_b :: s:cons_a:cons_b -> s:cons_a:cons_b hole_s:cons_a:cons_b1_0 :: s:cons_a:cons_b gen_s:cons_a:cons_b2_0 :: Nat -> s:cons_a:cons_b Generator Equations: gen_s:cons_a:cons_b2_0(0) <=> hole_s:cons_a:cons_b1_0 gen_s:cons_a:cons_b2_0(+(x, 1)) <=> s(gen_s:cons_a:cons_b2_0(x)) The following defined symbols remain to be analysed: b, a, encArg They will be analysed ascendingly in the following order: a = b a < encArg b < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_s:cons_a:cons_b2_0(+(1, n18_0))) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_s:cons_a:cons_b2_0(+(1, 0))) Induction Step: encArg(gen_s:cons_a:cons_b2_0(+(1, +(n18_0, 1)))) ->_R^Omega(0) s(encArg(gen_s:cons_a:cons_b2_0(+(1, n18_0)))) ->_IH s(*3_0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (12) BOUNDS(1, INF) ---------------------------------------- (13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (14) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(s(s(x1)))) -> s(s(a(a(x1)))) b(b(a(a(b(b(s(s(x1)))))))) -> a(a(b(b(s(s(a(a(x1)))))))) b(b(a(a(b(b(b(b(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) a(a(b(b(a(a(a(a(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (15) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (16) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(s(s(x1)))) -> s(s(a(a(x1)))) b(b(a(a(b(b(s(s(x1)))))))) -> a(a(b(b(s(s(a(a(x1)))))))) b(b(a(a(b(b(b(b(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) a(a(b(b(a(a(a(a(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (17) NonCtorToCtorProof (UPPER BOUND(ID)) transformed non-ctor to ctor-system ---------------------------------------- (18) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(c_a(s(s(x1)))) -> s(s(a(a(x1)))) a(c_a(c_b(c_b(c_a(c_a(c_a(c_a(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) b(c_b(c_a(c_a(c_b(c_b(s(s(x1)))))))) -> a(a(b(b(s(s(a(a(x1)))))))) b(c_b(c_a(c_a(c_b(c_b(c_b(c_b(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) a(x0) -> c_a(x0) b(x0) -> c_b(x0) Rewrite Strategy: INNERMOST ---------------------------------------- (19) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (20) 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: a(c_a(s(s(x1)))) -> s(s(a(a(x1)))) [1] a(c_a(c_b(c_b(c_a(c_a(c_a(c_a(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) [1] b(c_b(c_a(c_a(c_b(c_b(s(s(x1)))))))) -> a(a(b(b(s(s(a(a(x1)))))))) [1] b(c_b(c_a(c_a(c_b(c_b(c_b(c_b(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) [1] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_b(x_1) -> b(encArg(x_1)) [0] a(x0) -> c_a(x0) [0] b(x0) -> c_b(x0) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (21) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (22) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(c_a(s(s(x1)))) -> s(s(a(a(x1)))) [1] a(c_a(c_b(c_b(c_a(c_a(c_a(c_a(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) [1] b(c_b(c_a(c_a(c_b(c_b(s(s(x1)))))))) -> a(a(b(b(s(s(a(a(x1)))))))) [1] b(c_b(c_a(c_a(c_b(c_b(c_b(c_b(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) [1] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_b(x_1) -> b(encArg(x_1)) [0] a(x0) -> c_a(x0) [0] b(x0) -> c_b(x0) [0] The TRS has the following type information: a :: s:c_a:c_b:cons_a:cons_b -> s:c_a:c_b:cons_a:cons_b c_a :: s:c_a:c_b:cons_a:cons_b -> s:c_a:c_b:cons_a:cons_b s :: s:c_a:c_b:cons_a:cons_b -> s:c_a:c_b:cons_a:cons_b c_b :: s:c_a:c_b:cons_a:cons_b -> s:c_a:c_b:cons_a:cons_b b :: s:c_a:c_b:cons_a:cons_b -> s:c_a:c_b:cons_a:cons_b encArg :: s:c_a:c_b:cons_a:cons_b -> s:c_a:c_b:cons_a:cons_b cons_a :: s:c_a:c_b:cons_a:cons_b -> s:c_a:c_b:cons_a:cons_b cons_b :: s:c_a:c_b:cons_a:cons_b -> s:c_a:c_b:cons_a:cons_b encode_a :: s:c_a:c_b:cons_a:cons_b -> s:c_a:c_b:cons_a:cons_b encode_s :: s:c_a:c_b:cons_a:cons_b -> s:c_a:c_b:cons_a:cons_b encode_b :: s:c_a:c_b:cons_a:cons_b -> s:c_a:c_b:cons_a:cons_b Rewrite Strategy: INNERMOST ---------------------------------------- (23) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: encArg_1 encode_a_1 encode_s_1 encode_b_1 a_1 b_1 Due to the following rules being added: encArg(v0) -> const [0] encode_a(v0) -> const [0] encode_s(v0) -> const [0] encode_b(v0) -> const [0] a(v0) -> const [0] b(v0) -> const [0] And the following fresh constants: const ---------------------------------------- (24) 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: a(c_a(s(s(x1)))) -> s(s(a(a(x1)))) [1] a(c_a(c_b(c_b(c_a(c_a(c_a(c_a(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) [1] b(c_b(c_a(c_a(c_b(c_b(s(s(x1)))))))) -> a(a(b(b(s(s(a(a(x1)))))))) [1] b(c_b(c_a(c_a(c_b(c_b(c_b(c_b(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) [1] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_b(x_1) -> b(encArg(x_1)) [0] a(x0) -> c_a(x0) [0] b(x0) -> c_b(x0) [0] encArg(v0) -> const [0] encode_a(v0) -> const [0] encode_s(v0) -> const [0] encode_b(v0) -> const [0] a(v0) -> const [0] b(v0) -> const [0] The TRS has the following type information: a :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const c_a :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const s :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const c_b :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const b :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const encArg :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const cons_a :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const cons_b :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const encode_a :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const encode_s :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const encode_b :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const const :: s:c_a:c_b:cons_a:cons_b:const Rewrite Strategy: INNERMOST ---------------------------------------- (25) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (26) 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: a(c_a(s(s(c_a(s(s(x1'))))))) -> s(s(a(s(s(a(a(x1'))))))) [2] a(c_a(s(s(c_a(c_b(c_b(c_a(c_a(c_a(c_a(x1''))))))))))) -> s(s(a(b(b(a(a(b(b(a(a(x1''))))))))))) [2] a(c_a(s(s(x1)))) -> s(s(a(c_a(x1)))) [1] a(c_a(s(s(x1)))) -> s(s(a(const))) [1] a(c_a(c_b(c_b(c_a(c_a(c_a(c_a(c_a(s(s(x11))))))))))) -> b(b(a(a(b(b(a(s(s(a(a(x11))))))))))) [2] a(c_a(c_b(c_b(c_a(c_a(c_a(c_a(c_a(c_b(c_b(c_a(c_a(c_a(c_a(x12))))))))))))))) -> b(b(a(a(b(b(a(b(b(a(a(b(b(a(a(x12))))))))))))))) [2] a(c_a(c_b(c_b(c_a(c_a(c_a(c_a(x1)))))))) -> b(b(a(a(b(b(a(c_a(x1)))))))) [1] a(c_a(c_b(c_b(c_a(c_a(c_a(c_a(x1)))))))) -> b(b(a(a(b(b(a(const))))))) [1] b(c_b(c_a(c_a(c_b(c_b(s(s(c_a(s(s(x13))))))))))) -> a(a(b(b(s(s(a(s(s(a(a(x13))))))))))) [2] b(c_b(c_a(c_a(c_b(c_b(s(s(c_a(c_b(c_b(c_a(c_a(c_a(c_a(x14))))))))))))))) -> a(a(b(b(s(s(a(b(b(a(a(b(b(a(a(x14))))))))))))))) [2] b(c_b(c_a(c_a(c_b(c_b(s(s(x1)))))))) -> a(a(b(b(s(s(a(c_a(x1)))))))) [1] b(c_b(c_a(c_a(c_b(c_b(s(s(x1)))))))) -> a(a(b(b(s(s(a(const))))))) [1] b(c_b(c_a(c_a(c_b(c_b(c_b(c_b(c_b(c_a(c_a(c_b(c_b(s(s(x15))))))))))))))) -> a(a(b(b(a(a(b(a(a(b(b(s(s(a(a(x15))))))))))))))) [2] b(c_b(c_a(c_a(c_b(c_b(c_b(c_b(c_b(c_a(c_a(c_b(c_b(c_b(c_b(x16))))))))))))))) -> a(a(b(b(a(a(b(a(a(b(b(a(a(b(b(x16))))))))))))))) [2] b(c_b(c_a(c_a(c_b(c_b(c_b(c_b(x1)))))))) -> a(a(b(b(a(a(b(c_b(x1)))))))) [1] b(c_b(c_a(c_a(c_b(c_b(c_b(c_b(x1)))))))) -> a(a(b(b(a(a(b(const))))))) [1] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(cons_a(s(x_1'))) -> a(s(encArg(x_1'))) [0] encArg(cons_a(cons_a(x_1''))) -> a(a(encArg(x_1''))) [0] encArg(cons_a(cons_b(x_11))) -> a(b(encArg(x_11))) [0] encArg(cons_a(x_1)) -> a(const) [0] encArg(cons_b(s(x_12))) -> b(s(encArg(x_12))) [0] encArg(cons_b(cons_a(x_13))) -> b(a(encArg(x_13))) [0] encArg(cons_b(cons_b(x_14))) -> b(b(encArg(x_14))) [0] encArg(cons_b(x_1)) -> b(const) [0] encode_a(s(x_15)) -> a(s(encArg(x_15))) [0] encode_a(cons_a(x_16)) -> a(a(encArg(x_16))) [0] encode_a(cons_b(x_17)) -> a(b(encArg(x_17))) [0] encode_a(x_1) -> a(const) [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_b(s(x_18)) -> b(s(encArg(x_18))) [0] encode_b(cons_a(x_19)) -> b(a(encArg(x_19))) [0] encode_b(cons_b(x_110)) -> b(b(encArg(x_110))) [0] encode_b(x_1) -> b(const) [0] a(x0) -> c_a(x0) [0] b(x0) -> c_b(x0) [0] encArg(v0) -> const [0] encode_a(v0) -> const [0] encode_s(v0) -> const [0] encode_b(v0) -> const [0] a(v0) -> const [0] b(v0) -> const [0] The TRS has the following type information: a :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const c_a :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const s :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const c_b :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const b :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const encArg :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const cons_a :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const cons_b :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const encode_a :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const encode_s :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const encode_b :: s:c_a:c_b:cons_a:cons_b:const -> s:c_a:c_b:cons_a:cons_b:const const :: s:c_a:c_b:cons_a:cons_b:const Rewrite Strategy: INNERMOST ---------------------------------------- (27) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: const => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 2 }-> b(b(a(a(b(b(a(b(b(a(a(b(b(a(a(x12))))))))))))))) :|: z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + x12))))))))))))), x12 >= 0 a(z) -{ 1 }-> b(b(a(a(b(b(a(0))))))) :|: x1 >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + x1)))))) a(z) -{ 1 }-> b(b(a(a(b(b(a(1 + x1))))))) :|: x1 >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + x1)))))) a(z) -{ 2 }-> b(b(a(a(b(b(a(1 + (1 + a(a(x11)))))))))) :|: z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + x11))))))))), x11 >= 0 a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 a(z) -{ 2 }-> 1 + (1 + a(b(b(a(a(b(b(a(a(x1'')))))))))) :|: x1'' >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + x1''))))))))) a(z) -{ 1 }-> 1 + (1 + a(0)) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) a(z) -{ 1 }-> 1 + (1 + a(1 + x1)) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) a(z) -{ 2 }-> 1 + (1 + a(1 + (1 + a(a(x1'))))) :|: z = 1 + (1 + (1 + (1 + (1 + (1 + x1'))))), x1' >= 0 b(z) -{ 2 }-> a(a(b(b(a(a(b(a(a(b(b(a(a(b(b(x16))))))))))))))) :|: z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + x16))))))))))))), x16 >= 0 b(z) -{ 2 }-> a(a(b(b(a(a(b(a(a(b(b(1 + (1 + a(a(x15)))))))))))))) :|: z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + x15))))))))))))), x15 >= 0 b(z) -{ 1 }-> a(a(b(b(a(a(b(0))))))) :|: x1 >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + x1)))))) b(z) -{ 1 }-> a(a(b(b(a(a(b(1 + x1))))))) :|: x1 >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + x1)))))) b(z) -{ 2 }-> a(a(b(b(1 + (1 + a(b(b(a(a(b(b(a(a(x14)))))))))))))) :|: z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + x14))))))))))))), x14 >= 0 b(z) -{ 1 }-> a(a(b(b(1 + (1 + a(0)))))) :|: x1 >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + x1)))))) b(z) -{ 1 }-> a(a(b(b(1 + (1 + a(1 + x1)))))) :|: x1 >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + x1)))))) b(z) -{ 2 }-> a(a(b(b(1 + (1 + a(1 + (1 + a(a(x13))))))))) :|: x13 >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + x13))))))))) b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 b(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 encArg(z) -{ 0 }-> b(b(encArg(x_14))) :|: x_14 >= 0, z = 1 + (1 + x_14) encArg(z) -{ 0 }-> b(a(encArg(x_13))) :|: z = 1 + (1 + x_13), x_13 >= 0 encArg(z) -{ 0 }-> b(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> b(1 + encArg(x_12)) :|: z = 1 + (1 + x_12), x_12 >= 0 encArg(z) -{ 0 }-> a(b(encArg(x_11))) :|: x_11 >= 0, z = 1 + (1 + x_11) encArg(z) -{ 0 }-> a(a(encArg(x_1''))) :|: z = 1 + (1 + x_1''), x_1'' >= 0 encArg(z) -{ 0 }-> a(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> a(1 + encArg(x_1')) :|: z = 1 + (1 + x_1'), x_1' >= 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0 encode_a(z) -{ 0 }-> a(b(encArg(x_17))) :|: x_17 >= 0, z = 1 + x_17 encode_a(z) -{ 0 }-> a(a(encArg(x_16))) :|: z = 1 + x_16, x_16 >= 0 encode_a(z) -{ 0 }-> a(0) :|: x_1 >= 0, z = x_1 encode_a(z) -{ 0 }-> a(1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + x_15 encode_a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_b(z) -{ 0 }-> b(b(encArg(x_110))) :|: z = 1 + x_110, x_110 >= 0 encode_b(z) -{ 0 }-> b(a(encArg(x_19))) :|: z = 1 + x_19, x_19 >= 0 encode_b(z) -{ 0 }-> b(0) :|: x_1 >= 0, z = x_1 encode_b(z) -{ 0 }-> b(1 + encArg(x_18)) :|: z = 1 + x_18, x_18 >= 0 encode_b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1 ---------------------------------------- (29) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 2 }-> b(b(a(a(b(b(a(b(b(a(a(b(b(a(a(z - 14))))))))))))))) :|: z - 14 >= 0 a(z) -{ 1 }-> b(b(a(a(b(b(a(0))))))) :|: z - 7 >= 0 a(z) -{ 1 }-> b(b(a(a(b(b(a(1 + (z - 7)))))))) :|: z - 7 >= 0 a(z) -{ 2 }-> b(b(a(a(b(b(a(1 + (1 + a(a(z - 10)))))))))) :|: z - 10 >= 0 a(z) -{ 0 }-> 0 :|: z >= 0 a(z) -{ 0 }-> 1 + z :|: z >= 0 a(z) -{ 2 }-> 1 + (1 + a(b(b(a(a(b(b(a(a(z - 10)))))))))) :|: z - 10 >= 0 a(z) -{ 1 }-> 1 + (1 + a(0)) :|: z - 3 >= 0 a(z) -{ 1 }-> 1 + (1 + a(1 + (z - 3))) :|: z - 3 >= 0 a(z) -{ 2 }-> 1 + (1 + a(1 + (1 + a(a(z - 6))))) :|: z - 6 >= 0 b(z) -{ 2 }-> a(a(b(b(a(a(b(a(a(b(b(a(a(b(b(z - 14))))))))))))))) :|: z - 14 >= 0 b(z) -{ 2 }-> a(a(b(b(a(a(b(a(a(b(b(1 + (1 + a(a(z - 14)))))))))))))) :|: z - 14 >= 0 b(z) -{ 1 }-> a(a(b(b(a(a(b(0))))))) :|: z - 7 >= 0 b(z) -{ 1 }-> a(a(b(b(a(a(b(1 + (z - 7)))))))) :|: z - 7 >= 0 b(z) -{ 2 }-> a(a(b(b(1 + (1 + a(b(b(a(a(b(b(a(a(z - 14)))))))))))))) :|: z - 14 >= 0 b(z) -{ 1 }-> a(a(b(b(1 + (1 + a(0)))))) :|: z - 7 >= 0 b(z) -{ 1 }-> a(a(b(b(1 + (1 + a(1 + (z - 7))))))) :|: z - 7 >= 0 b(z) -{ 2 }-> a(a(b(b(1 + (1 + a(1 + (1 + a(a(z - 10))))))))) :|: z - 10 >= 0 b(z) -{ 0 }-> 0 :|: z >= 0 b(z) -{ 0 }-> 1 + z :|: z >= 0 encArg(z) -{ 0 }-> b(b(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> b(a(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> b(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> b(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> a(b(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> a(a(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> a(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> a(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_a(z) -{ 0 }-> a(b(encArg(z - 1))) :|: z - 1 >= 0 encode_a(z) -{ 0 }-> a(a(encArg(z - 1))) :|: z - 1 >= 0 encode_a(z) -{ 0 }-> a(0) :|: z >= 0 encode_a(z) -{ 0 }-> a(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_a(z) -{ 0 }-> 0 :|: z >= 0 encode_b(z) -{ 0 }-> b(b(encArg(z - 1))) :|: z - 1 >= 0 encode_b(z) -{ 0 }-> b(a(encArg(z - 1))) :|: z - 1 >= 0 encode_b(z) -{ 0 }-> b(0) :|: z >= 0 encode_b(z) -{ 0 }-> b(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_b(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 ---------------------------------------- (31) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: encArg(v0) -> null_encArg [0] encode_a(v0) -> null_encode_a [0] encode_s(v0) -> null_encode_s [0] encode_b(v0) -> null_encode_b [0] a(v0) -> null_a [0] b(v0) -> null_b [0] And the following fresh constants: null_encArg, null_encode_a, null_encode_s, null_encode_b, null_a, null_b ---------------------------------------- (32) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(c_a(s(s(x1)))) -> s(s(a(a(x1)))) [1] a(c_a(c_b(c_b(c_a(c_a(c_a(c_a(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) [1] b(c_b(c_a(c_a(c_b(c_b(s(s(x1)))))))) -> a(a(b(b(s(s(a(a(x1)))))))) [1] b(c_b(c_a(c_a(c_b(c_b(c_b(c_b(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) [1] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_b(x_1) -> b(encArg(x_1)) [0] a(x0) -> c_a(x0) [0] b(x0) -> c_b(x0) [0] encArg(v0) -> null_encArg [0] encode_a(v0) -> null_encode_a [0] encode_s(v0) -> null_encode_s [0] encode_b(v0) -> null_encode_b [0] a(v0) -> null_a [0] b(v0) -> null_b [0] The TRS has the following type information: a :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b -> s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b c_a :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b -> s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b s :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b -> s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b c_b :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b -> s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b b :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b -> s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b encArg :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b -> s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b cons_a :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b -> s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b cons_b :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b -> s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b encode_a :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b -> s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b encode_s :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b -> s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b encode_b :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b -> s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b null_encArg :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b null_encode_a :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b null_encode_s :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b null_encode_b :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b null_a :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b null_b :: s:c_a:c_b:cons_a:cons_b:null_encArg:null_encode_a:null_encode_s:null_encode_b:null_a:null_b Rewrite Strategy: INNERMOST ---------------------------------------- (33) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: null_encArg => 0 null_encode_a => 0 null_encode_s => 0 null_encode_b => 0 null_a => 0 null_b => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 1 }-> b(b(a(a(b(b(a(a(x1)))))))) :|: x1 >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + x1)))))) a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 a(z) -{ 1 }-> 1 + (1 + a(a(x1))) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) b(z) -{ 1 }-> a(a(b(b(a(a(b(b(x1)))))))) :|: x1 >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + x1)))))) b(z) -{ 1 }-> a(a(b(b(1 + (1 + a(a(x1))))))) :|: x1 >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + x1)))))) b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 b(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 encArg(z) -{ 0 }-> b(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> a(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0 encode_a(z) -{ 0 }-> a(encArg(x_1)) :|: x_1 >= 0, z = x_1 encode_a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_b(z) -{ 0 }-> b(encArg(x_1)) :|: x_1 >= 0, z = x_1 encode_b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) encode_s(z0) -> s(encArg(z0)) encode_b(z0) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_b(z0)) -> c2(B(encArg(z0)), ENCARG(z0)) ENCODE_A(z0) -> c3(A(encArg(z0)), ENCARG(z0)) ENCODE_S(z0) -> c4(ENCARG(z0)) ENCODE_B(z0) -> c5(B(encArg(z0)), ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(b(b(s(s(a(a(z0)))))))), A(b(b(s(s(a(a(z0))))))), B(b(s(s(a(a(z0)))))), B(s(s(a(a(z0))))), A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), A(b(b(z0))), B(b(z0)), B(z0)) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(b(b(s(s(a(a(z0)))))))), A(b(b(s(s(a(a(z0))))))), B(b(s(s(a(a(z0)))))), B(s(s(a(a(z0))))), A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), A(b(b(z0))), B(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: a_1, b_1, encArg_1, encode_a_1, encode_s_1, encode_b_1 Defined Pair Symbols: ENCARG_1, ENCODE_A_1, ENCODE_S_1, ENCODE_B_1, A_1, B_1 Compound Symbols: c_1, c1_2, c2_2, c3_2, c4_1, c5_2, c6_2, c7_8, c8_6, c9_8 ---------------------------------------- (37) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ENCODE_S(z0) -> c4(ENCARG(z0)) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) encode_s(z0) -> s(encArg(z0)) encode_b(z0) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_b(z0)) -> c2(B(encArg(z0)), ENCARG(z0)) ENCODE_A(z0) -> c3(A(encArg(z0)), ENCARG(z0)) ENCODE_B(z0) -> c5(B(encArg(z0)), ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(b(b(s(s(a(a(z0)))))))), A(b(b(s(s(a(a(z0))))))), B(b(s(s(a(a(z0)))))), B(s(s(a(a(z0))))), A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), A(b(b(z0))), B(b(z0)), B(z0)) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), B(a(a(z0))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(b(b(s(s(a(a(z0)))))))), A(b(b(s(s(a(a(z0))))))), B(b(s(s(a(a(z0)))))), B(s(s(a(a(z0))))), A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), A(b(b(z0))), B(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: a_1, b_1, encArg_1, encode_a_1, encode_s_1, encode_b_1 Defined Pair Symbols: ENCARG_1, ENCODE_A_1, ENCODE_B_1, A_1, B_1 Compound Symbols: c_1, c1_2, c2_2, c3_2, c5_2, c6_2, c7_8, c8_6, c9_8 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) encode_s(z0) -> s(encArg(z0)) encode_b(z0) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_b(z0)) -> c2(B(encArg(z0)), ENCARG(z0)) ENCODE_A(z0) -> c3(A(encArg(z0)), ENCARG(z0)) ENCODE_B(z0) -> c5(B(encArg(z0)), ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: a_1, b_1, encArg_1, encode_a_1, encode_s_1, encode_b_1 Defined Pair Symbols: ENCARG_1, ENCODE_A_1, ENCODE_B_1, A_1, B_1 Compound Symbols: c_1, c1_2, c2_2, c3_2, c5_2, c6_2, c7_7, c8_2, c9_7 ---------------------------------------- (41) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) encode_s(z0) -> s(encArg(z0)) encode_b(z0) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_b(z0)) -> c2(B(encArg(z0)), ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) ENCODE_A(z0) -> c4(A(encArg(z0))) ENCODE_A(z0) -> c4(ENCARG(z0)) ENCODE_B(z0) -> c4(B(encArg(z0))) ENCODE_B(z0) -> c4(ENCARG(z0)) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: a_1, b_1, encArg_1, encode_a_1, encode_s_1, encode_b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c1_2, c2_2, c6_2, c7_7, c8_2, c9_7, c4_1 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ENCODE_A(z0) -> c4(ENCARG(z0)) ENCODE_B(z0) -> c4(ENCARG(z0)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) encode_s(z0) -> s(encArg(z0)) encode_b(z0) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_b(z0)) -> c2(B(encArg(z0)), ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) ENCODE_A(z0) -> c4(A(encArg(z0))) ENCODE_B(z0) -> c4(B(encArg(z0))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: a_1, b_1, encArg_1, encode_a_1, encode_s_1, encode_b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c1_2, c2_2, c6_2, c7_7, c8_2, c9_7, c4_1 ---------------------------------------- (45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_a(z0) -> a(encArg(z0)) encode_s(z0) -> s(encArg(z0)) encode_b(z0) -> b(encArg(z0)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_b(z0)) -> c2(B(encArg(z0)), ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) ENCODE_A(z0) -> c4(A(encArg(z0))) ENCODE_B(z0) -> c4(B(encArg(z0))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c1_2, c2_2, c6_2, c7_7, c8_2, c9_7, c4_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) by ENCARG(cons_a(s(z0))) -> c1(A(s(encArg(z0))), ENCARG(s(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_b(z0)) -> c2(B(encArg(z0)), ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) ENCODE_A(z0) -> c4(A(encArg(z0))) ENCODE_B(z0) -> c4(B(encArg(z0))) ENCARG(cons_a(s(z0))) -> c1(A(s(encArg(z0))), ENCARG(s(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c2_2, c6_2, c7_7, c8_2, c9_7, c4_1, c1_2 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_b(z0)) -> c2(B(encArg(z0)), ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) ENCODE_A(z0) -> c4(A(encArg(z0))) ENCODE_B(z0) -> c4(B(encArg(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c2_2, c6_2, c7_7, c8_2, c9_7, c4_1, c1_2, c1_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(z0)) -> c2(B(encArg(z0)), ENCARG(z0)) by ENCARG(cons_b(s(z0))) -> c2(B(s(encArg(z0))), ENCARG(s(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) ENCODE_A(z0) -> c4(A(encArg(z0))) ENCODE_B(z0) -> c4(B(encArg(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(B(s(encArg(z0))), ENCARG(s(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c7_7, c8_2, c9_7, c4_1, c1_2, c1_1, c2_2 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) ENCODE_A(z0) -> c4(A(encArg(z0))) ENCODE_B(z0) -> c4(B(encArg(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c7_7, c8_2, c9_7, c4_1, c1_2, c1_1, c2_2, c2_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(a(b(b(a(a(a(a(z0)))))))) -> c7(B(b(a(a(b(b(a(a(z0)))))))), B(a(a(b(b(a(a(z0))))))), A(a(b(b(a(a(z0)))))), A(b(b(a(a(z0))))), B(b(a(a(z0)))), A(a(z0)), A(z0)) by A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0)))))))), A(b(b(a(a(a(a(z0))))))), B(b(a(a(a(a(z0)))))), A(a(a(a(z0)))), A(a(a(z0)))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(s(s(a(a(z0)))))))))))), B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0)))))))), A(a(b(b(s(s(z0)))))), A(b(b(s(s(z0)))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0)))))))), A(a(b(b(b(b(z0)))))), A(b(b(b(b(z0)))))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) ENCODE_A(z0) -> c4(A(encArg(z0))) ENCODE_B(z0) -> c4(B(encArg(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0)))))))), A(b(b(a(a(a(a(z0))))))), B(b(a(a(a(a(z0)))))), A(a(a(a(z0)))), A(a(a(z0)))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(s(s(a(a(z0)))))))))))), B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0)))))))), A(a(b(b(s(s(z0)))))), A(b(b(s(s(z0)))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0)))))))), A(a(b(b(b(b(z0)))))), A(b(b(b(b(z0)))))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0)))))))), A(b(b(a(a(a(a(z0))))))), B(b(a(a(a(a(z0)))))), A(a(a(a(z0)))), A(a(a(z0)))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(s(s(a(a(z0)))))))))))), B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0)))))))), A(a(b(b(s(s(z0)))))), A(b(b(s(s(z0)))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0)))))))), A(a(b(b(b(b(z0)))))), A(b(b(b(b(z0)))))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c9_7, c4_1, c1_2, c1_1, c2_2, c2_1, c7_7, c7_2 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing tuple parts ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) ENCODE_A(z0) -> c4(A(encArg(z0))) ENCODE_B(z0) -> c4(B(encArg(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c9_7, c4_1, c1_2, c1_1, c2_2, c2_1, c7_2, c7_3, c7_4, c7_5 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(b(a(a(b(b(b(b(z0)))))))) -> c9(A(a(b(b(a(a(b(b(z0)))))))), A(b(b(a(a(b(b(z0))))))), B(b(a(a(b(b(z0)))))), B(a(a(b(b(z0))))), A(a(b(b(z0)))), B(b(z0)), B(z0)) by B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(a(a(a(b(b(s(s(a(a(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0)))))))), B(a(a(b(b(s(s(z0))))))), A(a(b(b(s(s(z0)))))), B(b(s(s(z0)))), B(s(s(z0)))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0)))))))), B(a(a(b(b(b(b(z0))))))), A(a(b(b(b(b(z0)))))), B(b(b(b(z0)))), B(b(b(z0)))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0)))))))), B(b(a(a(a(a(z0)))))), B(a(a(a(a(z0)))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCODE_A(z0) -> c4(A(encArg(z0))) ENCODE_B(z0) -> c4(B(encArg(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(a(a(a(b(b(s(s(a(a(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0)))))))), B(a(a(b(b(s(s(z0))))))), A(a(b(b(s(s(z0)))))), B(b(s(s(z0)))), B(s(s(z0)))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0)))))))), B(a(a(b(b(b(b(z0))))))), A(a(b(b(b(b(z0)))))), B(b(b(b(z0)))), B(b(b(z0)))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0)))))))), B(b(a(a(a(a(z0)))))), B(a(a(a(a(z0)))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(a(a(a(b(b(s(s(a(a(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0)))))))), B(a(a(b(b(s(s(z0))))))), A(a(b(b(s(s(z0)))))), B(b(s(s(z0)))), B(s(s(z0)))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0)))))))), B(a(a(b(b(b(b(z0))))))), A(a(b(b(b(b(z0)))))), B(b(b(b(z0)))), B(b(b(z0)))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0)))))))), B(b(a(a(a(a(z0)))))), B(a(a(a(a(z0)))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c4_1, c1_2, c1_1, c2_2, c2_1, c7_2, c7_3, c7_4, c7_5, c9_7, c9_2 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCODE_A(z0) -> c4(A(encArg(z0))) ENCODE_B(z0) -> c4(B(encArg(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c4_1, c1_2, c1_1, c2_2, c2_1, c7_2, c7_3, c7_4, c7_5, c9_2, c9_3, c9_5 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_A(z0) -> c4(A(encArg(z0))) by ENCODE_A(s(z0)) -> c4(A(s(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCODE_B(z0) -> c4(B(encArg(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(s(z0)) -> c4(A(s(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c_1, c6_2, c8_2, c4_1, c1_2, c1_1, c2_2, c2_1, c7_2, c7_3, c7_4, c7_5, c9_2, c9_3, c9_5 ---------------------------------------- (65) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_A(s(z0)) -> c4(A(s(encArg(z0)))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCODE_B(z0) -> c4(B(encArg(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c_1, c6_2, c8_2, c4_1, c1_2, c1_1, c2_2, c2_1, c7_2, c7_3, c7_4, c7_5, c9_2, c9_3, c9_5 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_B(z0) -> c4(B(encArg(z0))) by ENCODE_B(s(z0)) -> c4(B(s(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(s(z0)) -> c4(B(s(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_2, c1_1, c2_2, c2_1, c7_2, c7_3, c7_4, c7_5, c9_2, c9_3, c9_5, c4_1 ---------------------------------------- (69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_B(s(z0)) -> c4(B(s(encArg(z0)))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_2, c1_1, c2_2, c2_1, c7_2, c7_3, c7_4, c7_5, c9_2, c9_3, c9_5, c4_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) by ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_2, c1_1, c2_2, c2_1, c7_2, c7_3, c7_4, c7_5, c9_2, c9_3, c9_5, c4_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) by ENCARG(cons_a(cons_b(s(z0)))) -> c1(A(b(s(encArg(z0)))), ENCARG(cons_b(s(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(A(b(s(encArg(z0)))), ENCARG(cons_b(s(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_2, c2_1, c7_2, c7_3, c7_4, c7_5, c9_2, c9_3, c9_5, c4_1, c1_2 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_2, c2_1, c7_2, c7_3, c7_4, c7_5, c9_2, c9_3, c9_5, c4_1, c1_2 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) by ENCARG(cons_b(cons_a(s(z0)))) -> c2(B(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(B(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_2, c2_1, c7_2, c7_3, c7_4, c7_5, c9_2, c9_3, c9_5, c4_1, c1_2 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_2, c2_1, c7_2, c7_3, c7_4, c7_5, c9_2, c9_3, c9_5, c4_1, c1_2 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) by ENCARG(cons_b(cons_b(s(z0)))) -> c2(B(b(s(encArg(z0)))), ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(B(b(s(encArg(z0)))), ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c7_3, c7_4, c7_5, c9_2, c9_3, c9_5, c4_1, c1_2, c2_2 ---------------------------------------- (83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c7_3, c7_4, c7_5, c9_2, c9_3, c9_5, c4_1, c1_2, c2_2 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(a(b(b(a(a(a(a(a(a(z0)))))))))) -> c7(B(b(b(b(a(a(b(b(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) by A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(s(s(a(a(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(s(s(a(a(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(s(s(a(a(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c7_4, c7_5, c9_2, c9_3, c9_5, c4_1, c1_2, c2_2, c7_3 ---------------------------------------- (87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c7_4, c7_5, c9_2, c9_3, c9_5, c4_1, c1_2, c2_2, c7_3 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(s(s(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) by A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(a(a(b(b(s(s(a(a(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(a(a(b(b(s(s(a(a(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(B(a(a(a(a(b(b(s(s(a(a(z0))))))))))), A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c7_5, c9_2, c9_3, c9_5, c4_1, c1_2, c2_2, c7_3, c7_4 ---------------------------------------- (91) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c7_5, c9_2, c9_3, c9_5, c4_1, c1_2, c2_2, c7_3 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))) -> c7(B(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), B(a(a(b(b(a(a(b(b(b(b(z0))))))))))), A(a(b(b(a(a(b(b(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) by A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(s(s(a(a(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(s(s(a(a(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(s(s(a(a(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c9_3, c9_5, c4_1, c1_2, c2_2, c7_3, c7_5, c7_4 ---------------------------------------- (95) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c9_3, c9_5, c4_1, c1_2, c2_2, c7_3, c7_5, c7_4 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) by B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(a(a(b(b(s(s(a(a(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(a(a(b(b(s(s(a(a(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(A(a(a(b(b(s(s(a(a(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c9_3, c9_5, c4_1, c1_2, c2_2, c7_3, c7_5, c7_4 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c9_3, c9_5, c4_1, c1_2, c2_2, c7_3, c7_5, c7_4, c9_1 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(b(a(a(b(b(b(b(b(b(z0)))))))))) -> c9(A(a(a(a(b(b(a(a(b(b(z0)))))))))), A(b(b(a(a(b(b(b(b(z0))))))))), B(b(a(a(b(b(b(b(z0))))))))) by B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(s(s(a(a(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(s(s(a(a(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(s(s(a(a(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c9_5, c4_1, c1_2, c2_2, c7_3, c7_5, c7_4, c9_1, c9_3 ---------------------------------------- (103) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c9_5, c4_1, c1_2, c2_2, c7_3, c7_5, c7_4, c9_1, c9_3 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))) -> c9(A(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), A(b(b(a(a(b(b(a(a(a(a(z0))))))))))), B(b(a(a(b(b(a(a(a(a(z0)))))))))), B(a(a(b(b(a(a(a(a(z0))))))))), A(a(b(b(a(a(a(a(z0))))))))) by B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(s(s(a(a(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(s(s(a(a(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(s(s(a(a(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c4_1, c1_2, c2_2, c7_3, c7_5, c7_4, c9_1, c9_3, c9_5, c9_4 ---------------------------------------- (107) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c4_1, c1_2, c2_2, c7_3, c7_5, c7_4, c9_1, c9_3, c9_5, c9_4 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_A(cons_a(z0)) -> c4(A(a(encArg(z0)))) by ENCODE_A(cons_a(s(z0))) -> c4(A(a(s(encArg(z0))))) ENCODE_A(cons_a(cons_a(z0))) -> c4(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_b(z0))) -> c4(A(a(b(encArg(z0))))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) ENCODE_A(cons_a(s(z0))) -> c4(A(a(s(encArg(z0))))) ENCODE_A(cons_a(cons_a(z0))) -> c4(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_b(z0))) -> c4(A(a(b(encArg(z0))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c4_1, c1_2, c2_2, c7_3, c7_5, c7_4, c9_1, c9_3, c9_5, c9_4 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_A(cons_b(z0)) -> c4(A(b(encArg(z0)))) by ENCODE_A(cons_b(s(z0))) -> c4(A(b(s(encArg(z0))))) ENCODE_A(cons_b(cons_a(z0))) -> c4(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c4(A(b(b(encArg(z0))))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) ENCODE_A(cons_a(s(z0))) -> c4(A(a(s(encArg(z0))))) ENCODE_A(cons_a(cons_a(z0))) -> c4(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_b(z0))) -> c4(A(a(b(encArg(z0))))) ENCODE_A(cons_b(s(z0))) -> c4(A(b(s(encArg(z0))))) ENCODE_A(cons_b(cons_a(z0))) -> c4(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c4(A(b(b(encArg(z0))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c4_1, c1_2, c2_2, c7_3, c7_5, c7_4, c9_1, c9_3, c9_5, c9_4 ---------------------------------------- (113) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_A(cons_b(s(z0))) -> c4(A(b(s(encArg(z0))))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) ENCODE_A(cons_a(s(z0))) -> c4(A(a(s(encArg(z0))))) ENCODE_A(cons_a(cons_a(z0))) -> c4(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_b(z0))) -> c4(A(a(b(encArg(z0))))) ENCODE_A(cons_b(cons_a(z0))) -> c4(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c4(A(b(b(encArg(z0))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c4_1, c1_2, c2_2, c7_3, c7_5, c7_4, c9_1, c9_3, c9_5, c9_4 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_B(cons_a(z0)) -> c4(B(a(encArg(z0)))) by ENCODE_B(cons_a(s(z0))) -> c4(B(a(s(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c4(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c4(B(a(b(encArg(z0))))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) ENCODE_A(cons_a(s(z0))) -> c4(A(a(s(encArg(z0))))) ENCODE_A(cons_a(cons_a(z0))) -> c4(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_b(z0))) -> c4(A(a(b(encArg(z0))))) ENCODE_A(cons_b(cons_a(z0))) -> c4(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c4(A(b(b(encArg(z0))))) ENCODE_B(cons_a(s(z0))) -> c4(B(a(s(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c4(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c4(B(a(b(encArg(z0))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c4_1, c1_2, c2_2, c7_3, c7_5, c7_4, c9_1, c9_3, c9_5, c9_4 ---------------------------------------- (117) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_B(cons_a(s(z0))) -> c4(B(a(s(encArg(z0))))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) ENCODE_A(cons_a(s(z0))) -> c4(A(a(s(encArg(z0))))) ENCODE_A(cons_a(cons_a(z0))) -> c4(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_b(z0))) -> c4(A(a(b(encArg(z0))))) ENCODE_A(cons_b(cons_a(z0))) -> c4(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c4(A(b(b(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c4(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c4(B(a(b(encArg(z0))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c4_1, c1_2, c2_2, c7_3, c7_5, c7_4, c9_1, c9_3, c9_5, c9_4 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_B(cons_b(z0)) -> c4(B(b(encArg(z0)))) by ENCODE_B(cons_b(s(z0))) -> c4(B(b(s(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c4(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c4(B(b(b(encArg(z0))))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) ENCODE_A(cons_a(s(z0))) -> c4(A(a(s(encArg(z0))))) ENCODE_A(cons_a(cons_a(z0))) -> c4(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_b(z0))) -> c4(A(a(b(encArg(z0))))) ENCODE_A(cons_b(cons_a(z0))) -> c4(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c4(A(b(b(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c4(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c4(B(a(b(encArg(z0))))) ENCODE_B(cons_b(s(z0))) -> c4(B(b(s(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c4(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c4(B(b(b(encArg(z0))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c1_2, c2_2, c7_3, c7_5, c7_4, c9_1, c9_3, c9_5, c9_4, c4_1 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_B(cons_b(s(z0))) -> c4(B(b(s(encArg(z0))))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) ENCODE_A(cons_a(s(z0))) -> c4(A(a(s(encArg(z0))))) ENCODE_A(cons_a(cons_a(z0))) -> c4(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_b(z0))) -> c4(A(a(b(encArg(z0))))) ENCODE_A(cons_b(cons_a(z0))) -> c4(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c4(A(b(b(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c4(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c4(B(a(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c4(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c4(B(b(b(encArg(z0))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c_1, c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c1_2, c2_2, c7_3, c7_5, c7_4, c9_1, c9_3, c9_5, c9_4, c4_1 ---------------------------------------- (123) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(s(z0)) -> c(ENCARG(z0)) by ENCARG(s(s(y0))) -> c(ENCARG(s(y0))) ENCARG(s(cons_a(s(y0)))) -> c(ENCARG(cons_a(s(y0)))) ENCARG(s(cons_b(s(y0)))) -> c(ENCARG(cons_b(s(y0)))) ENCARG(s(cons_a(cons_a(s(y0))))) -> c(ENCARG(cons_a(cons_a(s(y0))))) ENCARG(s(cons_a(cons_a(cons_a(y0))))) -> c(ENCARG(cons_a(cons_a(cons_a(y0))))) ENCARG(s(cons_a(cons_a(cons_b(y0))))) -> c(ENCARG(cons_a(cons_a(cons_b(y0))))) ENCARG(s(cons_a(cons_b(cons_a(y0))))) -> c(ENCARG(cons_a(cons_b(cons_a(y0))))) ENCARG(s(cons_a(cons_b(cons_b(y0))))) -> c(ENCARG(cons_a(cons_b(cons_b(y0))))) ENCARG(s(cons_a(cons_b(s(y0))))) -> c(ENCARG(cons_a(cons_b(s(y0))))) ENCARG(s(cons_b(cons_a(cons_a(y0))))) -> c(ENCARG(cons_b(cons_a(cons_a(y0))))) ENCARG(s(cons_b(cons_a(cons_b(y0))))) -> c(ENCARG(cons_b(cons_a(cons_b(y0))))) ENCARG(s(cons_b(cons_a(s(y0))))) -> c(ENCARG(cons_b(cons_a(s(y0))))) ENCARG(s(cons_b(cons_b(cons_a(y0))))) -> c(ENCARG(cons_b(cons_b(cons_a(y0))))) ENCARG(s(cons_b(cons_b(cons_b(y0))))) -> c(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(s(cons_b(cons_b(s(y0))))) -> c(ENCARG(cons_b(cons_b(s(y0))))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) ENCODE_A(cons_a(s(z0))) -> c4(A(a(s(encArg(z0))))) ENCODE_A(cons_a(cons_a(z0))) -> c4(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_b(z0))) -> c4(A(a(b(encArg(z0))))) ENCODE_A(cons_b(cons_a(z0))) -> c4(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c4(A(b(b(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c4(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c4(B(a(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c4(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c4(B(b(b(encArg(z0))))) ENCARG(s(s(y0))) -> c(ENCARG(s(y0))) ENCARG(s(cons_a(s(y0)))) -> c(ENCARG(cons_a(s(y0)))) ENCARG(s(cons_b(s(y0)))) -> c(ENCARG(cons_b(s(y0)))) ENCARG(s(cons_a(cons_a(s(y0))))) -> c(ENCARG(cons_a(cons_a(s(y0))))) ENCARG(s(cons_a(cons_a(cons_a(y0))))) -> c(ENCARG(cons_a(cons_a(cons_a(y0))))) ENCARG(s(cons_a(cons_a(cons_b(y0))))) -> c(ENCARG(cons_a(cons_a(cons_b(y0))))) ENCARG(s(cons_a(cons_b(cons_a(y0))))) -> c(ENCARG(cons_a(cons_b(cons_a(y0))))) ENCARG(s(cons_a(cons_b(cons_b(y0))))) -> c(ENCARG(cons_a(cons_b(cons_b(y0))))) ENCARG(s(cons_a(cons_b(s(y0))))) -> c(ENCARG(cons_a(cons_b(s(y0))))) ENCARG(s(cons_b(cons_a(cons_a(y0))))) -> c(ENCARG(cons_b(cons_a(cons_a(y0))))) ENCARG(s(cons_b(cons_a(cons_b(y0))))) -> c(ENCARG(cons_b(cons_a(cons_b(y0))))) ENCARG(s(cons_b(cons_a(s(y0))))) -> c(ENCARG(cons_b(cons_a(s(y0))))) ENCARG(s(cons_b(cons_b(cons_a(y0))))) -> c(ENCARG(cons_b(cons_b(cons_a(y0))))) ENCARG(s(cons_b(cons_b(cons_b(y0))))) -> c(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(s(cons_b(cons_b(s(y0))))) -> c(ENCARG(cons_b(cons_b(s(y0))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: A_1, B_1, ENCARG_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c6_2, c8_2, c1_1, c2_1, c7_2, c9_2, c1_2, c2_2, c7_3, c7_5, c7_4, c9_1, c9_3, c9_5, c9_4, c4_1, c_1 ---------------------------------------- (125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_a(s(z0))) -> c1(ENCARG(s(z0))) by ENCARG(cons_a(s(s(y0)))) -> c1(ENCARG(s(s(y0)))) ENCARG(cons_a(s(cons_a(s(y0))))) -> c1(ENCARG(s(cons_a(s(y0))))) ENCARG(cons_a(s(cons_b(s(y0))))) -> c1(ENCARG(s(cons_b(s(y0))))) ENCARG(cons_a(s(cons_a(cons_a(s(y0)))))) -> c1(ENCARG(s(cons_a(cons_a(s(y0)))))) ENCARG(cons_a(s(cons_a(cons_a(cons_a(y0)))))) -> c1(ENCARG(s(cons_a(cons_a(cons_a(y0)))))) ENCARG(cons_a(s(cons_a(cons_a(cons_b(y0)))))) -> c1(ENCARG(s(cons_a(cons_a(cons_b(y0)))))) ENCARG(cons_a(s(cons_a(cons_b(cons_a(y0)))))) -> c1(ENCARG(s(cons_a(cons_b(cons_a(y0)))))) ENCARG(cons_a(s(cons_a(cons_b(cons_b(y0)))))) -> c1(ENCARG(s(cons_a(cons_b(cons_b(y0)))))) ENCARG(cons_a(s(cons_a(cons_b(s(y0)))))) -> c1(ENCARG(s(cons_a(cons_b(s(y0)))))) ENCARG(cons_a(s(cons_b(cons_a(cons_a(y0)))))) -> c1(ENCARG(s(cons_b(cons_a(cons_a(y0)))))) ENCARG(cons_a(s(cons_b(cons_a(cons_b(y0)))))) -> c1(ENCARG(s(cons_b(cons_a(cons_b(y0)))))) ENCARG(cons_a(s(cons_b(cons_a(s(y0)))))) -> c1(ENCARG(s(cons_b(cons_a(s(y0)))))) ENCARG(cons_a(s(cons_b(cons_b(cons_a(y0)))))) -> c1(ENCARG(s(cons_b(cons_b(cons_a(y0)))))) ENCARG(cons_a(s(cons_b(cons_b(cons_b(y0)))))) -> c1(ENCARG(s(cons_b(cons_b(cons_b(y0)))))) ENCARG(cons_a(s(cons_b(cons_b(s(y0)))))) -> c1(ENCARG(s(cons_b(cons_b(s(y0)))))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) ENCODE_A(cons_a(s(z0))) -> c4(A(a(s(encArg(z0))))) ENCODE_A(cons_a(cons_a(z0))) -> c4(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_b(z0))) -> c4(A(a(b(encArg(z0))))) ENCODE_A(cons_b(cons_a(z0))) -> c4(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c4(A(b(b(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c4(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c4(B(a(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c4(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c4(B(b(b(encArg(z0))))) ENCARG(s(s(y0))) -> c(ENCARG(s(y0))) ENCARG(s(cons_a(s(y0)))) -> c(ENCARG(cons_a(s(y0)))) ENCARG(s(cons_b(s(y0)))) -> c(ENCARG(cons_b(s(y0)))) ENCARG(s(cons_a(cons_a(s(y0))))) -> c(ENCARG(cons_a(cons_a(s(y0))))) ENCARG(s(cons_a(cons_a(cons_a(y0))))) -> c(ENCARG(cons_a(cons_a(cons_a(y0))))) ENCARG(s(cons_a(cons_a(cons_b(y0))))) -> c(ENCARG(cons_a(cons_a(cons_b(y0))))) ENCARG(s(cons_a(cons_b(cons_a(y0))))) -> c(ENCARG(cons_a(cons_b(cons_a(y0))))) ENCARG(s(cons_a(cons_b(cons_b(y0))))) -> c(ENCARG(cons_a(cons_b(cons_b(y0))))) ENCARG(s(cons_a(cons_b(s(y0))))) -> c(ENCARG(cons_a(cons_b(s(y0))))) ENCARG(s(cons_b(cons_a(cons_a(y0))))) -> c(ENCARG(cons_b(cons_a(cons_a(y0))))) ENCARG(s(cons_b(cons_a(cons_b(y0))))) -> c(ENCARG(cons_b(cons_a(cons_b(y0))))) ENCARG(s(cons_b(cons_a(s(y0))))) -> c(ENCARG(cons_b(cons_a(s(y0))))) ENCARG(s(cons_b(cons_b(cons_a(y0))))) -> c(ENCARG(cons_b(cons_b(cons_a(y0))))) ENCARG(s(cons_b(cons_b(cons_b(y0))))) -> c(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(s(cons_b(cons_b(s(y0))))) -> c(ENCARG(cons_b(cons_b(s(y0))))) ENCARG(cons_a(s(s(y0)))) -> c1(ENCARG(s(s(y0)))) ENCARG(cons_a(s(cons_a(s(y0))))) -> c1(ENCARG(s(cons_a(s(y0))))) ENCARG(cons_a(s(cons_b(s(y0))))) -> c1(ENCARG(s(cons_b(s(y0))))) ENCARG(cons_a(s(cons_a(cons_a(s(y0)))))) -> c1(ENCARG(s(cons_a(cons_a(s(y0)))))) ENCARG(cons_a(s(cons_a(cons_a(cons_a(y0)))))) -> c1(ENCARG(s(cons_a(cons_a(cons_a(y0)))))) ENCARG(cons_a(s(cons_a(cons_a(cons_b(y0)))))) -> c1(ENCARG(s(cons_a(cons_a(cons_b(y0)))))) ENCARG(cons_a(s(cons_a(cons_b(cons_a(y0)))))) -> c1(ENCARG(s(cons_a(cons_b(cons_a(y0)))))) ENCARG(cons_a(s(cons_a(cons_b(cons_b(y0)))))) -> c1(ENCARG(s(cons_a(cons_b(cons_b(y0)))))) ENCARG(cons_a(s(cons_a(cons_b(s(y0)))))) -> c1(ENCARG(s(cons_a(cons_b(s(y0)))))) ENCARG(cons_a(s(cons_b(cons_a(cons_a(y0)))))) -> c1(ENCARG(s(cons_b(cons_a(cons_a(y0)))))) ENCARG(cons_a(s(cons_b(cons_a(cons_b(y0)))))) -> c1(ENCARG(s(cons_b(cons_a(cons_b(y0)))))) ENCARG(cons_a(s(cons_b(cons_a(s(y0)))))) -> c1(ENCARG(s(cons_b(cons_a(s(y0)))))) ENCARG(cons_a(s(cons_b(cons_b(cons_a(y0)))))) -> c1(ENCARG(s(cons_b(cons_b(cons_a(y0)))))) ENCARG(cons_a(s(cons_b(cons_b(cons_b(y0)))))) -> c1(ENCARG(s(cons_b(cons_b(cons_b(y0)))))) ENCARG(cons_a(s(cons_b(cons_b(s(y0)))))) -> c1(ENCARG(s(cons_b(cons_b(s(y0)))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: A_1, B_1, ENCARG_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c6_2, c8_2, c2_1, c7_2, c9_2, c1_2, c1_1, c2_2, c7_3, c7_5, c7_4, c9_1, c9_3, c9_5, c9_4, c4_1, c_1 ---------------------------------------- (127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_b(s(z0))) -> c2(ENCARG(s(z0))) by ENCARG(cons_b(s(s(y0)))) -> c2(ENCARG(s(s(y0)))) ENCARG(cons_b(s(cons_a(s(y0))))) -> c2(ENCARG(s(cons_a(s(y0))))) ENCARG(cons_b(s(cons_b(s(y0))))) -> c2(ENCARG(s(cons_b(s(y0))))) ENCARG(cons_b(s(cons_a(cons_a(s(y0)))))) -> c2(ENCARG(s(cons_a(cons_a(s(y0)))))) ENCARG(cons_b(s(cons_a(cons_a(cons_a(y0)))))) -> c2(ENCARG(s(cons_a(cons_a(cons_a(y0)))))) ENCARG(cons_b(s(cons_a(cons_a(cons_b(y0)))))) -> c2(ENCARG(s(cons_a(cons_a(cons_b(y0)))))) ENCARG(cons_b(s(cons_a(cons_b(cons_a(y0)))))) -> c2(ENCARG(s(cons_a(cons_b(cons_a(y0)))))) ENCARG(cons_b(s(cons_a(cons_b(cons_b(y0)))))) -> c2(ENCARG(s(cons_a(cons_b(cons_b(y0)))))) ENCARG(cons_b(s(cons_a(cons_b(s(y0)))))) -> c2(ENCARG(s(cons_a(cons_b(s(y0)))))) ENCARG(cons_b(s(cons_b(cons_a(cons_a(y0)))))) -> c2(ENCARG(s(cons_b(cons_a(cons_a(y0)))))) ENCARG(cons_b(s(cons_b(cons_a(cons_b(y0)))))) -> c2(ENCARG(s(cons_b(cons_a(cons_b(y0)))))) ENCARG(cons_b(s(cons_b(cons_a(s(y0)))))) -> c2(ENCARG(s(cons_b(cons_a(s(y0)))))) ENCARG(cons_b(s(cons_b(cons_b(cons_a(y0)))))) -> c2(ENCARG(s(cons_b(cons_b(cons_a(y0)))))) ENCARG(cons_b(s(cons_b(cons_b(cons_b(y0)))))) -> c2(ENCARG(s(cons_b(cons_b(cons_b(y0)))))) ENCARG(cons_b(s(cons_b(cons_b(s(y0)))))) -> c2(ENCARG(s(cons_b(cons_b(s(y0)))))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(s(s(z0)))) -> s(s(a(a(z0)))) a(a(b(b(a(a(a(a(z0)))))))) -> b(b(a(a(b(b(a(a(z0)))))))) b(b(a(a(b(b(s(s(z0)))))))) -> a(a(b(b(s(s(a(a(z0)))))))) b(b(a(a(b(b(b(b(z0)))))))) -> a(a(b(b(a(a(b(b(z0)))))))) Tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) ENCARG(cons_a(cons_a(s(z0)))) -> c1(A(a(s(encArg(z0)))), ENCARG(cons_a(s(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(s(z0)))) -> c1(ENCARG(cons_b(s(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(s(z0)))) -> c2(ENCARG(cons_a(s(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(s(z0)))) -> c2(ENCARG(cons_b(s(z0)))) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) ENCODE_A(cons_a(s(z0))) -> c4(A(a(s(encArg(z0))))) ENCODE_A(cons_a(cons_a(z0))) -> c4(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_b(z0))) -> c4(A(a(b(encArg(z0))))) ENCODE_A(cons_b(cons_a(z0))) -> c4(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c4(A(b(b(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c4(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c4(B(a(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c4(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c4(B(b(b(encArg(z0))))) ENCARG(s(s(y0))) -> c(ENCARG(s(y0))) ENCARG(s(cons_a(s(y0)))) -> c(ENCARG(cons_a(s(y0)))) ENCARG(s(cons_b(s(y0)))) -> c(ENCARG(cons_b(s(y0)))) ENCARG(s(cons_a(cons_a(s(y0))))) -> c(ENCARG(cons_a(cons_a(s(y0))))) ENCARG(s(cons_a(cons_a(cons_a(y0))))) -> c(ENCARG(cons_a(cons_a(cons_a(y0))))) ENCARG(s(cons_a(cons_a(cons_b(y0))))) -> c(ENCARG(cons_a(cons_a(cons_b(y0))))) ENCARG(s(cons_a(cons_b(cons_a(y0))))) -> c(ENCARG(cons_a(cons_b(cons_a(y0))))) ENCARG(s(cons_a(cons_b(cons_b(y0))))) -> c(ENCARG(cons_a(cons_b(cons_b(y0))))) ENCARG(s(cons_a(cons_b(s(y0))))) -> c(ENCARG(cons_a(cons_b(s(y0))))) ENCARG(s(cons_b(cons_a(cons_a(y0))))) -> c(ENCARG(cons_b(cons_a(cons_a(y0))))) ENCARG(s(cons_b(cons_a(cons_b(y0))))) -> c(ENCARG(cons_b(cons_a(cons_b(y0))))) ENCARG(s(cons_b(cons_a(s(y0))))) -> c(ENCARG(cons_b(cons_a(s(y0))))) ENCARG(s(cons_b(cons_b(cons_a(y0))))) -> c(ENCARG(cons_b(cons_b(cons_a(y0))))) ENCARG(s(cons_b(cons_b(cons_b(y0))))) -> c(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(s(cons_b(cons_b(s(y0))))) -> c(ENCARG(cons_b(cons_b(s(y0))))) ENCARG(cons_a(s(s(y0)))) -> c1(ENCARG(s(s(y0)))) ENCARG(cons_a(s(cons_a(s(y0))))) -> c1(ENCARG(s(cons_a(s(y0))))) ENCARG(cons_a(s(cons_b(s(y0))))) -> c1(ENCARG(s(cons_b(s(y0))))) ENCARG(cons_a(s(cons_a(cons_a(s(y0)))))) -> c1(ENCARG(s(cons_a(cons_a(s(y0)))))) ENCARG(cons_a(s(cons_a(cons_a(cons_a(y0)))))) -> c1(ENCARG(s(cons_a(cons_a(cons_a(y0)))))) ENCARG(cons_a(s(cons_a(cons_a(cons_b(y0)))))) -> c1(ENCARG(s(cons_a(cons_a(cons_b(y0)))))) ENCARG(cons_a(s(cons_a(cons_b(cons_a(y0)))))) -> c1(ENCARG(s(cons_a(cons_b(cons_a(y0)))))) ENCARG(cons_a(s(cons_a(cons_b(cons_b(y0)))))) -> c1(ENCARG(s(cons_a(cons_b(cons_b(y0)))))) ENCARG(cons_a(s(cons_a(cons_b(s(y0)))))) -> c1(ENCARG(s(cons_a(cons_b(s(y0)))))) ENCARG(cons_a(s(cons_b(cons_a(cons_a(y0)))))) -> c1(ENCARG(s(cons_b(cons_a(cons_a(y0)))))) ENCARG(cons_a(s(cons_b(cons_a(cons_b(y0)))))) -> c1(ENCARG(s(cons_b(cons_a(cons_b(y0)))))) ENCARG(cons_a(s(cons_b(cons_a(s(y0)))))) -> c1(ENCARG(s(cons_b(cons_a(s(y0)))))) ENCARG(cons_a(s(cons_b(cons_b(cons_a(y0)))))) -> c1(ENCARG(s(cons_b(cons_b(cons_a(y0)))))) ENCARG(cons_a(s(cons_b(cons_b(cons_b(y0)))))) -> c1(ENCARG(s(cons_b(cons_b(cons_b(y0)))))) ENCARG(cons_a(s(cons_b(cons_b(s(y0)))))) -> c1(ENCARG(s(cons_b(cons_b(s(y0)))))) ENCARG(cons_b(s(s(y0)))) -> c2(ENCARG(s(s(y0)))) ENCARG(cons_b(s(cons_a(s(y0))))) -> c2(ENCARG(s(cons_a(s(y0))))) ENCARG(cons_b(s(cons_b(s(y0))))) -> c2(ENCARG(s(cons_b(s(y0))))) ENCARG(cons_b(s(cons_a(cons_a(s(y0)))))) -> c2(ENCARG(s(cons_a(cons_a(s(y0)))))) ENCARG(cons_b(s(cons_a(cons_a(cons_a(y0)))))) -> c2(ENCARG(s(cons_a(cons_a(cons_a(y0)))))) ENCARG(cons_b(s(cons_a(cons_a(cons_b(y0)))))) -> c2(ENCARG(s(cons_a(cons_a(cons_b(y0)))))) ENCARG(cons_b(s(cons_a(cons_b(cons_a(y0)))))) -> c2(ENCARG(s(cons_a(cons_b(cons_a(y0)))))) ENCARG(cons_b(s(cons_a(cons_b(cons_b(y0)))))) -> c2(ENCARG(s(cons_a(cons_b(cons_b(y0)))))) ENCARG(cons_b(s(cons_a(cons_b(s(y0)))))) -> c2(ENCARG(s(cons_a(cons_b(s(y0)))))) ENCARG(cons_b(s(cons_b(cons_a(cons_a(y0)))))) -> c2(ENCARG(s(cons_b(cons_a(cons_a(y0)))))) ENCARG(cons_b(s(cons_b(cons_a(cons_b(y0)))))) -> c2(ENCARG(s(cons_b(cons_a(cons_b(y0)))))) ENCARG(cons_b(s(cons_b(cons_a(s(y0)))))) -> c2(ENCARG(s(cons_b(cons_a(s(y0)))))) ENCARG(cons_b(s(cons_b(cons_b(cons_a(y0)))))) -> c2(ENCARG(s(cons_b(cons_b(cons_a(y0)))))) ENCARG(cons_b(s(cons_b(cons_b(cons_b(y0)))))) -> c2(ENCARG(s(cons_b(cons_b(cons_b(y0)))))) ENCARG(cons_b(s(cons_b(cons_b(s(y0)))))) -> c2(ENCARG(s(cons_b(cons_b(s(y0)))))) S tuples: A(a(s(s(z0)))) -> c6(A(a(z0)), A(z0)) B(b(a(a(b(b(s(s(z0)))))))) -> c8(A(a(z0)), A(z0)) A(a(b(b(a(a(a(a(x0)))))))) -> c7(A(a(x0)), A(x0)) B(b(a(a(b(b(b(b(x0)))))))) -> c9(B(b(x0)), B(x0)) A(a(b(b(a(a(a(a(a(a(a(a(z0)))))))))))) -> c7(B(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) A(a(b(b(a(a(a(a(a(a(b(b(b(b(z0)))))))))))))) -> c7(B(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) A(a(b(b(a(a(a(a(a(a(x0)))))))))) -> c7(B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) A(a(b(b(a(a(a(a(a(a(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))) -> c7(A(a(b(b(a(a(b(b(s(s(z0)))))))))), A(b(b(a(a(b(b(s(s(z0))))))))), B(b(a(a(b(b(s(s(z0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(b(b(z0)))))))))))))) -> c7(B(b(a(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))), B(a(a(b(b(a(a(b(b(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))))) -> c7(B(b(a(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))))), B(a(a(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))))), A(a(b(b(a(a(b(b(b(b(a(a(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) A(a(b(b(a(a(a(a(b(b(b(b(x0)))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(x0))))))))))), A(a(b(b(a(a(b(b(b(b(x0)))))))))), A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) A(a(b(b(a(a(a(a(b(b(b(b(s(s(z0)))))))))))))) -> c7(B(a(a(b(b(a(a(b(b(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(b(b(b(b(s(s(z0)))))))))))), A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(s(s(z0)))))))))) -> c9(B(b(a(a(b(b(s(s(z0))))))))) B(b(a(a(b(b(b(b(b(b(b(b(z0)))))))))))) -> c9(A(a(a(a(a(a(b(b(a(a(b(b(z0)))))))))))), A(b(b(a(a(b(b(b(b(b(b(z0))))))))))), B(b(a(a(b(b(b(b(b(b(z0))))))))))) B(b(a(a(b(b(b(b(b(b(a(a(a(a(z0)))))))))))))) -> c9(A(a(a(a(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(b(b(a(a(a(a(z0))))))))))))) B(b(a(a(b(b(b(b(b(b(x0)))))))))) -> c9(A(b(b(a(a(b(b(b(b(x0))))))))), B(b(a(a(b(b(b(b(x0))))))))) B(b(a(a(b(b(b(b(b(b(s(s(z0)))))))))))) -> c9(A(b(b(a(a(b(b(b(b(s(s(z0))))))))))), B(b(a(a(b(b(b(b(s(s(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(a(a(z0)))))))))))))) -> c9(A(a(b(b(b(b(b(b(a(a(b(b(a(a(z0)))))))))))))), A(b(b(a(a(b(b(a(a(a(a(a(a(z0))))))))))))), B(b(a(a(b(b(a(a(a(a(a(a(z0)))))))))))), B(a(a(b(b(a(a(a(a(a(a(z0))))))))))), A(a(b(b(a(a(a(a(a(a(z0))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))))) -> c9(A(a(b(b(b(b(a(a(a(a(b(b(a(a(b(b(z0)))))))))))))))), A(b(b(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(b(b(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(b(b(z0))))))))))))) B(b(a(a(b(b(b(b(a(a(a(a(x0)))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(x0))))))))))), B(b(a(a(b(b(a(a(a(a(x0)))))))))), B(a(a(b(b(a(a(a(a(x0))))))))), A(a(b(b(a(a(a(a(x0))))))))) B(b(a(a(b(b(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))))) -> c9(A(b(b(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))))), B(b(a(a(b(b(a(a(a(a(b(b(s(s(z0)))))))))))))), B(a(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))), A(a(b(b(a(a(a(a(b(b(s(s(z0))))))))))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: A_1, B_1, ENCARG_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c6_2, c8_2, c7_2, c9_2, c1_2, c1_1, c2_2, c2_1, c7_3, c7_5, c7_4, c9_1, c9_3, c9_5, c9_4, c4_1, c_1