/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- KILLED proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, INF).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), 161 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxRelTRS (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) typed CpxTrs (11) OrderProof [LOWER BOUND(ID), 0 ms] (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 608 ms] (14) BOUNDS(1, INF) (15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxTRS (17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (18) CpxRelTRS (19) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRelTRS (21) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxWeightedTrs (23) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxTypedWeightedTrs (25) CompletionProof [UPPER BOUND(ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) CompletionProof [UPPER BOUND(ID), 0 ms] (30) CpxTypedWeightedCompleteTrs (31) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CpxRNTS (37) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (38) CdtProblem (39) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (46) CdtProblem (47) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 12 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 7 ms] (150) CdtProblem (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 37 ms] (156) CdtProblem ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(f(b, a(x)))) -> f(a(a(a(x))), b) a(a(x)) -> f(b, a(f(a(x), b))) f(a(x), b) -> f(b, a(x)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (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(b) -> b encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_b -> b ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(f(b, a(x)))) -> f(a(a(a(x))), b) a(a(x)) -> f(b, a(f(a(x), b))) f(a(x), b) -> f(b, a(x)) The (relative) TRS S consists of the following rules: encArg(b) -> b encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_b -> b Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(f(b, a(x)))) -> f(a(a(a(x))), b) a(a(x)) -> f(b, a(f(a(x), b))) f(a(x), b) -> f(b, a(x)) The (relative) TRS S consists of the following rules: encArg(b) -> b encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_b -> b Rewrite Strategy: FULL ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(f(b, a(x)))) -> f(a(a(a(x))), b) a(a(x)) -> f(b, a(f(a(x), b))) f(a(x), b) -> f(b, a(x)) The (relative) TRS S consists of the following rules: encArg(b) -> b encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_b -> b Rewrite Strategy: FULL ---------------------------------------- (7) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (8) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(f(b, a(x)))) -> f(a(a(a(x))), b) a(a(x)) -> f(b, a(f(a(x), b))) f(a(x), b) -> f(b, a(x)) The (relative) TRS S consists of the following rules: encArg(b) -> b encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_b -> b Rewrite Strategy: FULL ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: TRS: Rules: a(a(f(b, a(x)))) -> f(a(a(a(x))), b) a(a(x)) -> f(b, a(f(a(x), b))) f(a(x), b) -> f(b, a(x)) encArg(b) -> b encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_b -> b Types: a :: b:cons_a:cons_f -> b:cons_a:cons_f f :: b:cons_a:cons_f -> b:cons_a:cons_f -> b:cons_a:cons_f b :: b:cons_a:cons_f encArg :: b:cons_a:cons_f -> b:cons_a:cons_f cons_a :: b:cons_a:cons_f -> b:cons_a:cons_f cons_f :: b:cons_a:cons_f -> b:cons_a:cons_f -> b:cons_a:cons_f encode_a :: b:cons_a:cons_f -> b:cons_a:cons_f encode_f :: b:cons_a:cons_f -> b:cons_a:cons_f -> b:cons_a:cons_f encode_b :: b:cons_a:cons_f hole_b:cons_a:cons_f1_0 :: b:cons_a:cons_f gen_b:cons_a:cons_f2_0 :: Nat -> b:cons_a:cons_f ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: a, f, encArg They will be analysed ascendingly in the following order: a = f a < encArg f < encArg ---------------------------------------- (12) Obligation: TRS: Rules: a(a(f(b, a(x)))) -> f(a(a(a(x))), b) a(a(x)) -> f(b, a(f(a(x), b))) f(a(x), b) -> f(b, a(x)) encArg(b) -> b encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_b -> b Types: a :: b:cons_a:cons_f -> b:cons_a:cons_f f :: b:cons_a:cons_f -> b:cons_a:cons_f -> b:cons_a:cons_f b :: b:cons_a:cons_f encArg :: b:cons_a:cons_f -> b:cons_a:cons_f cons_a :: b:cons_a:cons_f -> b:cons_a:cons_f cons_f :: b:cons_a:cons_f -> b:cons_a:cons_f -> b:cons_a:cons_f encode_a :: b:cons_a:cons_f -> b:cons_a:cons_f encode_f :: b:cons_a:cons_f -> b:cons_a:cons_f -> b:cons_a:cons_f encode_b :: b:cons_a:cons_f hole_b:cons_a:cons_f1_0 :: b:cons_a:cons_f gen_b:cons_a:cons_f2_0 :: Nat -> b:cons_a:cons_f Generator Equations: gen_b:cons_a:cons_f2_0(0) <=> b gen_b:cons_a:cons_f2_0(+(x, 1)) <=> cons_a(gen_b:cons_a:cons_f2_0(x)) The following defined symbols remain to be analysed: f, a, encArg They will be analysed ascendingly in the following order: a = f a < encArg f < encArg ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_b:cons_a:cons_f2_0(+(1, n15_0))) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_b:cons_a:cons_f2_0(+(1, 0))) Induction Step: encArg(gen_b:cons_a:cons_f2_0(+(1, +(n15_0, 1)))) ->_R^Omega(0) a(encArg(gen_b:cons_a:cons_f2_0(+(1, n15_0)))) ->_IH a(*3_0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (14) BOUNDS(1, INF) ---------------------------------------- (15) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (16) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(f(b, a(x)))) -> f(a(a(a(x))), b) a(a(x)) -> f(b, a(f(a(x), b))) f(a(x), b) -> f(b, a(x)) encArg(b) -> b encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_b -> b S is empty. Rewrite Strategy: FULL ---------------------------------------- (17) NonCtorToCtorProof (UPPER BOUND(ID)) transformed non-ctor to ctor-system ---------------------------------------- (18) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(c_a(c_f(b, c_a(x)))) -> f(a(a(a(x))), b) f(c_a(x), b) -> f(b, a(x)) a(c_a(x)) -> f(b, a(f(a(x), b))) The (relative) TRS S consists of the following rules: encArg(b) -> b encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_b -> b a(x0) -> c_a(x0) f(x0, x1) -> c_f(x0, x1) Rewrite Strategy: FULL ---------------------------------------- (19) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. As the TRS is a non-duplicating overlay system, we have rc = irc. ---------------------------------------- (20) 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(c_f(b, c_a(x)))) -> f(a(a(a(x))), b) f(c_a(x), b) -> f(b, a(x)) a(c_a(x)) -> f(b, a(f(a(x), b))) The (relative) TRS S consists of the following rules: encArg(b) -> b encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_b -> b a(x0) -> c_a(x0) f(x0, x1) -> c_f(x0, x1) Rewrite Strategy: INNERMOST ---------------------------------------- (21) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (22) 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(c_f(b, c_a(x)))) -> f(a(a(a(x))), b) [1] f(c_a(x), b) -> f(b, a(x)) [1] a(c_a(x)) -> f(b, a(f(a(x), b))) [1] encArg(b) -> b [0] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0] encode_b -> b [0] a(x0) -> c_a(x0) [0] f(x0, x1) -> c_f(x0, x1) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (23) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (24) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(c_a(c_f(b, c_a(x)))) -> f(a(a(a(x))), b) [1] f(c_a(x), b) -> f(b, a(x)) [1] a(c_a(x)) -> f(b, a(f(a(x), b))) [1] encArg(b) -> b [0] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0] encode_b -> b [0] a(x0) -> c_a(x0) [0] f(x0, x1) -> c_f(x0, x1) [0] The TRS has the following type information: a :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f c_a :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f c_f :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f b :: b:c_a:c_f:cons_a:cons_f f :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f encArg :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f cons_a :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f cons_f :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f encode_a :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f encode_f :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f encode_b :: b:c_a:c_f:cons_a:cons_f Rewrite Strategy: INNERMOST ---------------------------------------- (25) 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_f(v0, v1) -> null_encode_f [0] encode_b -> null_encode_b [0] a(v0) -> null_a [0] f(v0, v1) -> null_f [0] And the following fresh constants: null_encArg, null_encode_a, null_encode_f, null_encode_b, null_a, null_f ---------------------------------------- (26) 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(c_f(b, c_a(x)))) -> f(a(a(a(x))), b) [1] f(c_a(x), b) -> f(b, a(x)) [1] a(c_a(x)) -> f(b, a(f(a(x), b))) [1] encArg(b) -> b [0] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0] encode_b -> b [0] a(x0) -> c_a(x0) [0] f(x0, x1) -> c_f(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_a(v0) -> null_encode_a [0] encode_f(v0, v1) -> null_encode_f [0] encode_b -> null_encode_b [0] a(v0) -> null_a [0] f(v0, v1) -> null_f [0] The TRS has the following type information: a :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f -> b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f c_a :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f -> b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f c_f :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f -> b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f -> b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f b :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f f :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f -> b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f -> b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f encArg :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f -> b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f cons_a :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f -> b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f cons_f :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f -> b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f -> b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f encode_a :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f -> b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f encode_f :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f -> b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f -> b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f encode_b :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f null_encArg :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f null_encode_a :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f null_encode_f :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f null_encode_b :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f null_a :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f null_f :: b:c_a:c_f:cons_a:cons_f:null_encArg:null_encode_a:null_encode_f:null_encode_b:null_a:null_f 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: b => 0 null_encArg => 0 null_encode_a => 0 null_encode_f => 0 null_encode_b => 0 null_a => 0 null_f => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 1 }-> f(a(a(a(x))), 0) :|: z = 1 + (1 + 0 + (1 + x)), x >= 0 a(z) -{ 1 }-> f(0, a(f(a(x), 0))) :|: x >= 0, z = 1 + x a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> a(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 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 -{ 0 }-> 0 :|: encode_f(z, z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 f(z, z') -{ 1 }-> f(0, a(x)) :|: x >= 0, z = 1 + x, z' = 0 f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 f(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (29) 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_f_2 encode_b a_1 f_2 Due to the following rules being added: encArg(v0) -> b [0] encode_a(v0) -> b [0] encode_f(v0, v1) -> b [0] encode_b -> b [0] a(v0) -> b [0] f(v0, v1) -> b [0] And the following fresh constants: none ---------------------------------------- (30) 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(c_f(b, c_a(x)))) -> f(a(a(a(x))), b) [1] f(c_a(x), b) -> f(b, a(x)) [1] a(c_a(x)) -> f(b, a(f(a(x), b))) [1] encArg(b) -> b [0] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0] encode_b -> b [0] a(x0) -> c_a(x0) [0] f(x0, x1) -> c_f(x0, x1) [0] encArg(v0) -> b [0] encode_a(v0) -> b [0] encode_f(v0, v1) -> b [0] encode_b -> b [0] a(v0) -> b [0] f(v0, v1) -> b [0] The TRS has the following type information: a :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f c_a :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f c_f :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f b :: b:c_a:c_f:cons_a:cons_f f :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f encArg :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f cons_a :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f cons_f :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f encode_a :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f encode_f :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f encode_b :: b:c_a:c_f:cons_a:cons_f Rewrite Strategy: INNERMOST ---------------------------------------- (31) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (32) 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(c_f(b, c_a(c_a(c_f(b, c_a(x'))))))) -> f(a(a(f(a(a(a(x'))), b))), b) [2] a(c_a(c_f(b, c_a(c_a(x''))))) -> f(a(a(f(b, a(f(a(x''), b))))), b) [2] a(c_a(c_f(b, c_a(x)))) -> f(a(a(c_a(x))), b) [1] a(c_a(c_f(b, c_a(x)))) -> f(a(a(b)), b) [1] f(c_a(c_a(c_f(b, c_a(x2)))), b) -> f(b, f(a(a(a(x2))), b)) [2] f(c_a(c_a(x3)), b) -> f(b, f(b, a(f(a(x3), b)))) [2] f(c_a(x), b) -> f(b, c_a(x)) [1] f(c_a(x), b) -> f(b, b) [1] a(c_a(c_a(c_f(b, c_a(x4))))) -> f(b, a(f(f(a(a(a(x4))), b), b))) [2] a(c_a(c_a(x5))) -> f(b, a(f(f(b, a(f(a(x5), b))), b))) [2] a(c_a(x)) -> f(b, a(f(c_a(x), b))) [1] a(c_a(x)) -> f(b, a(f(b, b))) [1] encArg(b) -> b [0] encArg(cons_a(b)) -> a(b) [0] encArg(cons_a(cons_a(x_1'))) -> a(a(encArg(x_1'))) [0] encArg(cons_a(cons_f(x_1'', x_2'))) -> a(f(encArg(x_1''), encArg(x_2'))) [0] encArg(cons_a(x_1)) -> a(b) [0] encArg(cons_f(b, b)) -> f(b, b) [0] encArg(cons_f(b, cons_a(x_13))) -> f(b, a(encArg(x_13))) [0] encArg(cons_f(b, cons_f(x_14, x_21))) -> f(b, f(encArg(x_14), encArg(x_21))) [0] encArg(cons_f(b, x_2)) -> f(b, b) [0] encArg(cons_f(cons_a(x_11), b)) -> f(a(encArg(x_11)), b) [0] encArg(cons_f(cons_a(x_11), cons_a(x_15))) -> f(a(encArg(x_11)), a(encArg(x_15))) [0] encArg(cons_f(cons_a(x_11), cons_f(x_16, x_22))) -> f(a(encArg(x_11)), f(encArg(x_16), encArg(x_22))) [0] encArg(cons_f(cons_a(x_11), x_2)) -> f(a(encArg(x_11)), b) [0] encArg(cons_f(cons_f(x_12, x_2''), b)) -> f(f(encArg(x_12), encArg(x_2'')), b) [0] encArg(cons_f(cons_f(x_12, x_2''), cons_a(x_17))) -> f(f(encArg(x_12), encArg(x_2'')), a(encArg(x_17))) [0] encArg(cons_f(cons_f(x_12, x_2''), cons_f(x_18, x_23))) -> f(f(encArg(x_12), encArg(x_2'')), f(encArg(x_18), encArg(x_23))) [0] encArg(cons_f(cons_f(x_12, x_2''), x_2)) -> f(f(encArg(x_12), encArg(x_2'')), b) [0] encArg(cons_f(x_1, b)) -> f(b, b) [0] encArg(cons_f(x_1, cons_a(x_19))) -> f(b, a(encArg(x_19))) [0] encArg(cons_f(x_1, cons_f(x_110, x_24))) -> f(b, f(encArg(x_110), encArg(x_24))) [0] encArg(cons_f(x_1, x_2)) -> f(b, b) [0] encode_a(b) -> a(b) [0] encode_a(cons_a(x_111)) -> a(a(encArg(x_111))) [0] encode_a(cons_f(x_112, x_25)) -> a(f(encArg(x_112), encArg(x_25))) [0] encode_a(x_1) -> a(b) [0] encode_f(b, b) -> f(b, b) [0] encode_f(b, cons_a(x_115)) -> f(b, a(encArg(x_115))) [0] encode_f(b, cons_f(x_116, x_27)) -> f(b, f(encArg(x_116), encArg(x_27))) [0] encode_f(b, x_2) -> f(b, b) [0] encode_f(cons_a(x_113), b) -> f(a(encArg(x_113)), b) [0] encode_f(cons_a(x_113), cons_a(x_117)) -> f(a(encArg(x_113)), a(encArg(x_117))) [0] encode_f(cons_a(x_113), cons_f(x_118, x_28)) -> f(a(encArg(x_113)), f(encArg(x_118), encArg(x_28))) [0] encode_f(cons_a(x_113), x_2) -> f(a(encArg(x_113)), b) [0] encode_f(cons_f(x_114, x_26), b) -> f(f(encArg(x_114), encArg(x_26)), b) [0] encode_f(cons_f(x_114, x_26), cons_a(x_119)) -> f(f(encArg(x_114), encArg(x_26)), a(encArg(x_119))) [0] encode_f(cons_f(x_114, x_26), cons_f(x_120, x_29)) -> f(f(encArg(x_114), encArg(x_26)), f(encArg(x_120), encArg(x_29))) [0] encode_f(cons_f(x_114, x_26), x_2) -> f(f(encArg(x_114), encArg(x_26)), b) [0] encode_f(x_1, b) -> f(b, b) [0] encode_f(x_1, cons_a(x_121)) -> f(b, a(encArg(x_121))) [0] encode_f(x_1, cons_f(x_122, x_210)) -> f(b, f(encArg(x_122), encArg(x_210))) [0] encode_f(x_1, x_2) -> f(b, b) [0] encode_b -> b [0] a(x0) -> c_a(x0) [0] f(x0, x1) -> c_f(x0, x1) [0] encArg(v0) -> b [0] encode_a(v0) -> b [0] encode_f(v0, v1) -> b [0] encode_b -> b [0] a(v0) -> b [0] f(v0, v1) -> b [0] The TRS has the following type information: a :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f c_a :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f c_f :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f b :: b:c_a:c_f:cons_a:cons_f f :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f encArg :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f cons_a :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f cons_f :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f encode_a :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f encode_f :: b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f -> b:c_a:c_f:cons_a:cons_f encode_b :: b:c_a:c_f:cons_a:cons_f Rewrite Strategy: INNERMOST ---------------------------------------- (33) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: b => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 2 }-> f(a(a(f(a(a(a(x'))), 0))), 0) :|: x' >= 0, z = 1 + (1 + 0 + (1 + (1 + (1 + 0 + (1 + x'))))) a(z) -{ 2 }-> f(a(a(f(0, a(f(a(x''), 0))))), 0) :|: z = 1 + (1 + 0 + (1 + (1 + x''))), x'' >= 0 a(z) -{ 1 }-> f(a(a(0)), 0) :|: z = 1 + (1 + 0 + (1 + x)), x >= 0 a(z) -{ 1 }-> f(a(a(1 + x)), 0) :|: z = 1 + (1 + 0 + (1 + x)), x >= 0 a(z) -{ 2 }-> f(0, a(f(f(a(a(a(x4))), 0), 0))) :|: x4 >= 0, z = 1 + (1 + (1 + 0 + (1 + x4))) a(z) -{ 2 }-> f(0, a(f(f(0, a(f(a(x5), 0))), 0))) :|: x5 >= 0, z = 1 + (1 + x5) a(z) -{ 1 }-> f(0, a(f(0, 0))) :|: x >= 0, z = 1 + x a(z) -{ 1 }-> f(0, a(f(1 + x, 0))) :|: x >= 0, z = 1 + x a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 encArg(z) -{ 0 }-> f(f(encArg(x_12), encArg(x_2'')), f(encArg(x_18), encArg(x_23))) :|: x_2'' >= 0, x_23 >= 0, x_12 >= 0, x_18 >= 0, z = 1 + (1 + x_12 + x_2'') + (1 + x_18 + x_23) encArg(z) -{ 0 }-> f(f(encArg(x_12), encArg(x_2'')), a(encArg(x_17))) :|: z = 1 + (1 + x_12 + x_2'') + (1 + x_17), x_17 >= 0, x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> f(f(encArg(x_12), encArg(x_2'')), 0) :|: x_2'' >= 0, z = 1 + (1 + x_12 + x_2'') + 0, x_12 >= 0 encArg(z) -{ 0 }-> f(f(encArg(x_12), encArg(x_2'')), 0) :|: x_2'' >= 0, z = 1 + (1 + x_12 + x_2'') + x_2, x_2 >= 0, x_12 >= 0 encArg(z) -{ 0 }-> f(a(encArg(x_11)), f(encArg(x_16), encArg(x_22))) :|: x_11 >= 0, z = 1 + (1 + x_11) + (1 + x_16 + x_22), x_16 >= 0, x_22 >= 0 encArg(z) -{ 0 }-> f(a(encArg(x_11)), a(encArg(x_15))) :|: x_11 >= 0, x_15 >= 0, z = 1 + (1 + x_11) + (1 + x_15) encArg(z) -{ 0 }-> f(a(encArg(x_11)), 0) :|: x_11 >= 0, z = 1 + (1 + x_11) + 0 encArg(z) -{ 0 }-> f(a(encArg(x_11)), 0) :|: x_11 >= 0, z = 1 + (1 + x_11) + x_2, x_2 >= 0 encArg(z) -{ 0 }-> f(0, f(encArg(x_110), encArg(x_24))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_110 + x_24), x_24 >= 0, x_110 >= 0 encArg(z) -{ 0 }-> f(0, f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + 0 + (1 + x_14 + x_21), x_21 >= 0 encArg(z) -{ 0 }-> f(0, a(encArg(x_13))) :|: z = 1 + 0 + (1 + x_13), x_13 >= 0 encArg(z) -{ 0 }-> f(0, a(encArg(x_19))) :|: z = 1 + x_1 + (1 + x_19), x_1 >= 0, x_19 >= 0 encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + 0 encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + x_1 + 0, x_1 >= 0 encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> a(f(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> a(a(encArg(x_1'))) :|: z = 1 + (1 + x_1'), x_1' >= 0 encArg(z) -{ 0 }-> a(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> a(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_a(z) -{ 0 }-> a(f(encArg(x_112), encArg(x_25))) :|: z = 1 + x_112 + x_25, x_25 >= 0, x_112 >= 0 encode_a(z) -{ 0 }-> a(a(encArg(x_111))) :|: z = 1 + x_111, x_111 >= 0 encode_a(z) -{ 0 }-> a(0) :|: z = 0 encode_a(z) -{ 0 }-> a(0) :|: x_1 >= 0, z = x_1 encode_a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_b -{ 0 }-> 0 :|: encode_f(z, z') -{ 0 }-> f(f(encArg(x_114), encArg(x_26)), f(encArg(x_120), encArg(x_29))) :|: x_120 >= 0, z' = 1 + x_120 + x_29, x_114 >= 0, x_26 >= 0, z = 1 + x_114 + x_26, x_29 >= 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_114), encArg(x_26)), a(encArg(x_119))) :|: z' = 1 + x_119, x_114 >= 0, x_26 >= 0, z = 1 + x_114 + x_26, x_119 >= 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_114), encArg(x_26)), 0) :|: x_114 >= 0, x_26 >= 0, z = 1 + x_114 + x_26, z' = 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_114), encArg(x_26)), 0) :|: x_114 >= 0, x_26 >= 0, x_2 >= 0, z = 1 + x_114 + x_26, z' = x_2 encode_f(z, z') -{ 0 }-> f(a(encArg(x_113)), f(encArg(x_118), encArg(x_28))) :|: z = 1 + x_113, x_113 >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0 encode_f(z, z') -{ 0 }-> f(a(encArg(x_113)), a(encArg(x_117))) :|: z = 1 + x_113, x_113 >= 0, x_117 >= 0, z' = 1 + x_117 encode_f(z, z') -{ 0 }-> f(a(encArg(x_113)), 0) :|: z = 1 + x_113, x_113 >= 0, z' = 0 encode_f(z, z') -{ 0 }-> f(a(encArg(x_113)), 0) :|: z = 1 + x_113, x_113 >= 0, x_2 >= 0, z' = x_2 encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, x_27 >= 0, z = 0 encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_122), encArg(x_210))) :|: x_1 >= 0, x_122 >= 0, z' = 1 + x_122 + x_210, z = x_1, x_210 >= 0 encode_f(z, z') -{ 0 }-> f(0, a(encArg(x_115))) :|: x_115 >= 0, z' = 1 + x_115, z = 0 encode_f(z, z') -{ 0 }-> f(0, a(encArg(x_121))) :|: x_1 >= 0, z = x_1, z' = 1 + x_121, x_121 >= 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: z = 0, z' = 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: x_2 >= 0, z' = x_2, z = 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = x_1, z' = 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 f(z, z') -{ 2 }-> f(0, f(a(a(a(x2))), 0)) :|: z = 1 + (1 + (1 + 0 + (1 + x2))), x2 >= 0, z' = 0 f(z, z') -{ 2 }-> f(0, f(0, a(f(a(x3), 0)))) :|: z = 1 + (1 + x3), x3 >= 0, z' = 0 f(z, z') -{ 1 }-> f(0, 0) :|: x >= 0, z = 1 + x, z' = 0 f(z, z') -{ 1 }-> f(0, 1 + x) :|: x >= 0, z = 1 + x, z' = 0 f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 f(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 ---------------------------------------- (35) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 2 }-> f(a(a(f(a(a(a(z - 6))), 0))), 0) :|: z - 6 >= 0 a(z) -{ 2 }-> f(a(a(f(0, a(f(a(z - 4), 0))))), 0) :|: z - 4 >= 0 a(z) -{ 1 }-> f(a(a(0)), 0) :|: z - 3 >= 0 a(z) -{ 1 }-> f(a(a(1 + (z - 3))), 0) :|: z - 3 >= 0 a(z) -{ 2 }-> f(0, a(f(f(a(a(a(z - 4))), 0), 0))) :|: z - 4 >= 0 a(z) -{ 2 }-> f(0, a(f(f(0, a(f(a(z - 2), 0))), 0))) :|: z - 2 >= 0 a(z) -{ 1 }-> f(0, a(f(0, 0))) :|: z - 1 >= 0 a(z) -{ 1 }-> f(0, a(f(1 + (z - 1), 0))) :|: z - 1 >= 0 a(z) -{ 0 }-> 0 :|: z >= 0 a(z) -{ 0 }-> 1 + z :|: z >= 0 encArg(z) -{ 0 }-> f(f(encArg(x_12), encArg(x_2'')), f(encArg(x_18), encArg(x_23))) :|: x_2'' >= 0, x_23 >= 0, x_12 >= 0, x_18 >= 0, z = 1 + (1 + x_12 + x_2'') + (1 + x_18 + x_23) encArg(z) -{ 0 }-> f(f(encArg(x_12), encArg(x_2'')), a(encArg(x_17))) :|: z = 1 + (1 + x_12 + x_2'') + (1 + x_17), x_17 >= 0, x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> f(f(encArg(x_12), encArg(x_2'')), 0) :|: x_2'' >= 0, z = 1 + (1 + x_12 + x_2'') + 0, x_12 >= 0 encArg(z) -{ 0 }-> f(f(encArg(x_12), encArg(x_2'')), 0) :|: x_2'' >= 0, z = 1 + (1 + x_12 + x_2'') + x_2, x_2 >= 0, x_12 >= 0 encArg(z) -{ 0 }-> f(a(encArg(x_11)), f(encArg(x_16), encArg(x_22))) :|: x_11 >= 0, z = 1 + (1 + x_11) + (1 + x_16 + x_22), x_16 >= 0, x_22 >= 0 encArg(z) -{ 0 }-> f(a(encArg(x_11)), a(encArg(x_15))) :|: x_11 >= 0, x_15 >= 0, z = 1 + (1 + x_11) + (1 + x_15) encArg(z) -{ 0 }-> f(a(encArg(x_11)), 0) :|: x_11 >= 0, z = 1 + (1 + x_11) + x_2, x_2 >= 0 encArg(z) -{ 0 }-> f(a(encArg(z - 2)), 0) :|: z - 2 >= 0 encArg(z) -{ 0 }-> f(0, f(encArg(x_110), encArg(x_24))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_110 + x_24), x_24 >= 0, x_110 >= 0 encArg(z) -{ 0 }-> f(0, f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + 0 + (1 + x_14 + x_21), x_21 >= 0 encArg(z) -{ 0 }-> f(0, a(encArg(x_19))) :|: z = 1 + x_1 + (1 + x_19), x_1 >= 0, x_19 >= 0 encArg(z) -{ 0 }-> f(0, a(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + 0 encArg(z) -{ 0 }-> f(0, 0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> a(f(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_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(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_a(z) -{ 0 }-> a(f(encArg(x_112), encArg(x_25))) :|: z = 1 + x_112 + x_25, x_25 >= 0, x_112 >= 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(0) :|: z >= 0 encode_a(z) -{ 0 }-> 0 :|: z >= 0 encode_b -{ 0 }-> 0 :|: encode_f(z, z') -{ 0 }-> f(f(encArg(x_114), encArg(x_26)), f(encArg(x_120), encArg(x_29))) :|: x_120 >= 0, z' = 1 + x_120 + x_29, x_114 >= 0, x_26 >= 0, z = 1 + x_114 + x_26, x_29 >= 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_114), encArg(x_26)), a(encArg(z' - 1))) :|: x_114 >= 0, x_26 >= 0, z = 1 + x_114 + x_26, z' - 1 >= 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_114), encArg(x_26)), 0) :|: x_114 >= 0, x_26 >= 0, z = 1 + x_114 + x_26, z' = 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_114), encArg(x_26)), 0) :|: x_114 >= 0, x_26 >= 0, z' >= 0, z = 1 + x_114 + x_26 encode_f(z, z') -{ 0 }-> f(a(encArg(z - 1)), f(encArg(x_118), encArg(x_28))) :|: z - 1 >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0 encode_f(z, z') -{ 0 }-> f(a(encArg(z - 1)), a(encArg(z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0 encode_f(z, z') -{ 0 }-> f(a(encArg(z - 1)), 0) :|: z - 1 >= 0, z' = 0 encode_f(z, z') -{ 0 }-> f(a(encArg(z - 1)), 0) :|: z - 1 >= 0, z' >= 0 encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, x_27 >= 0, z = 0 encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_122), encArg(x_210))) :|: z >= 0, x_122 >= 0, z' = 1 + x_122 + x_210, x_210 >= 0 encode_f(z, z') -{ 0 }-> f(0, a(encArg(z' - 1))) :|: z' - 1 >= 0, z = 0 encode_f(z, z') -{ 0 }-> f(0, a(encArg(z' - 1))) :|: z >= 0, z' - 1 >= 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: z = 0, z' = 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: z' >= 0, z = 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' = 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' >= 0 encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 f(z, z') -{ 2 }-> f(0, f(a(a(a(z - 4))), 0)) :|: z - 4 >= 0, z' = 0 f(z, z') -{ 2 }-> f(0, f(0, a(f(a(z - 2), 0)))) :|: z - 2 >= 0, z' = 0 f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' = 0 f(z, z') -{ 1 }-> f(0, 1 + (z - 1)) :|: z - 1 >= 0, z' = 0 f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0 ---------------------------------------- (37) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_a(z0) -> a(encArg(z0)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_b -> b a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: ENCARG(b) -> c ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_A(z0) -> c3(A(encArg(z0)), ENCARG(z0)) ENCODE_F(z0, z1) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_B -> c5 A(z0) -> c6 A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(F(b, a(f(a(z0), b))), A(f(a(z0), b)), F(a(z0), b), A(z0)) F(z0, z1) -> c9 F(c_a(z0), b) -> c10(F(b, a(z0)), A(z0)) S tuples: A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(F(b, a(f(a(z0), b))), A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(F(b, a(z0)), A(z0)) K tuples:none Defined Rule Symbols: a_1, f_2, encArg_1, encode_a_1, encode_f_2, encode_b Defined Pair Symbols: ENCARG_1, ENCODE_A_1, ENCODE_F_2, ENCODE_B, A_1, F_2 Compound Symbols: c, c1_2, c2_3, c3_2, c4_3, c5, c6, c7_4, c8_4, c9, c10_2 ---------------------------------------- (39) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: ENCODE_B -> c5 A(z0) -> c6 F(z0, z1) -> c9 ENCARG(b) -> c ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_a(z0) -> a(encArg(z0)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_b -> b a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_A(z0) -> c3(A(encArg(z0)), ENCARG(z0)) ENCODE_F(z0, z1) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(F(b, a(f(a(z0), b))), A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(F(b, a(z0)), A(z0)) S tuples: A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(F(b, a(f(a(z0), b))), A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(F(b, a(z0)), A(z0)) K tuples:none Defined Rule Symbols: a_1, f_2, encArg_1, encode_a_1, encode_f_2, encode_b Defined Pair Symbols: ENCARG_1, ENCODE_A_1, ENCODE_F_2, A_1, F_2 Compound Symbols: c1_2, c2_3, c3_2, c4_3, c7_4, c8_4, c10_2 ---------------------------------------- (41) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_a(z0) -> a(encArg(z0)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_b -> b a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_A(z0) -> c3(A(encArg(z0)), ENCARG(z0)) ENCODE_F(z0, z1) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) S tuples: A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) K tuples:none Defined Rule Symbols: a_1, f_2, encArg_1, encode_a_1, encode_f_2, encode_b Defined Pair Symbols: ENCARG_1, ENCODE_A_1, ENCODE_F_2, A_1, F_2 Compound Symbols: c1_2, c2_3, c3_2, c4_3, c7_4, c8_3, c10_1 ---------------------------------------- (43) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_a(z0) -> a(encArg(z0)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_b -> b a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) ENCODE_A(z0) -> c(A(encArg(z0))) ENCODE_A(z0) -> c(ENCARG(z0)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCODE_F(z0, z1) -> c(ENCARG(z0)) ENCODE_F(z0, z1) -> c(ENCARG(z1)) S tuples: A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) K tuples:none Defined Rule Symbols: a_1, f_2, encArg_1, encode_a_1, encode_f_2, encode_b Defined Pair Symbols: ENCARG_1, A_1, F_2, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c1_2, c2_3, c7_4, c8_3, c10_1, c_1 ---------------------------------------- (45) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 3 leading nodes: ENCODE_A(z0) -> c(ENCARG(z0)) ENCODE_F(z0, z1) -> c(ENCARG(z0)) ENCODE_F(z0, z1) -> c(ENCARG(z1)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_a(z0) -> a(encArg(z0)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_b -> b a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) ENCODE_A(z0) -> c(A(encArg(z0))) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) S tuples: A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) K tuples:none Defined Rule Symbols: a_1, f_2, encArg_1, encode_a_1, encode_f_2, encode_b Defined Pair Symbols: ENCARG_1, A_1, F_2, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c1_2, c2_3, c7_4, c8_3, c10_1, c_1 ---------------------------------------- (47) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_a(z0) -> a(encArg(z0)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_b -> b ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) ENCODE_A(z0) -> c(A(encArg(z0))) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) S tuples: A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: ENCARG_1, A_1, F_2, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c1_2, c2_3, c7_4, c8_3, c10_1, c_1 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) by ENCARG(cons_a(b)) -> c1(A(b), ENCARG(b)) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_f(z0, z1))) -> c1(A(f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) ENCODE_A(z0) -> c(A(encArg(z0))) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_a(b)) -> c1(A(b), ENCARG(b)) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_f(z0, z1))) -> c1(A(f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) S tuples: A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: ENCARG_1, A_1, F_2, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c2_3, c7_4, c8_3, c10_1, c_1, c1_2 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCARG(cons_a(b)) -> c1(A(b), ENCARG(b)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) ENCODE_A(z0) -> c(A(encArg(z0))) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_f(z0, z1))) -> c1(A(f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) S tuples: A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: ENCARG_1, A_1, F_2, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c2_3, c7_4, c8_3, c10_1, c_1, c1_2 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0), ENCARG(b)) ENCARG(cons_f(x0, cons_a(z0))) -> c2(F(encArg(x0), a(encArg(z0))), ENCARG(x0), ENCARG(cons_a(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, x1)) -> c2(F(b, encArg(x1)), ENCARG(b), ENCARG(x1)) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) ENCODE_A(z0) -> c(A(encArg(z0))) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_f(z0, z1))) -> c1(A(f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0), ENCARG(b)) ENCARG(cons_f(x0, cons_a(z0))) -> c2(F(encArg(x0), a(encArg(z0))), ENCARG(x0), ENCARG(cons_a(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, x1)) -> c2(F(b, encArg(x1)), ENCARG(b), ENCARG(x1)) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) S tuples: A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: A_1, F_2, ENCODE_A_1, ENCODE_F_2, ENCARG_1 Compound Symbols: c7_4, c8_3, c10_1, c_1, c1_2, c2_3 ---------------------------------------- (55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) ENCODE_A(z0) -> c(A(encArg(z0))) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_f(z0, z1))) -> c1(A(f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, cons_a(z0))) -> c2(F(encArg(x0), a(encArg(z0))), ENCARG(x0), ENCARG(cons_a(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) S tuples: A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: A_1, F_2, ENCODE_A_1, ENCODE_F_2, ENCARG_1 Compound Symbols: c7_4, c8_3, c10_1, c_1, c1_2, c2_3, c2_2, c2_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) by A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) ENCODE_A(z0) -> c(A(encArg(z0))) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_f(z0, z1))) -> c1(A(f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, cons_a(z0))) -> c2(F(encArg(x0), a(encArg(z0))), ENCARG(x0), ENCARG(cons_a(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) S tuples: A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: A_1, F_2, ENCODE_A_1, ENCODE_F_2, ENCARG_1 Compound Symbols: c8_3, c10_1, c_1, c1_2, c2_3, c2_2, c2_1, c7_4 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(z0)) -> c8(A(f(a(z0), b)), F(a(z0), b), A(z0)) by A(c_a(x0)) -> c8(A(c_f(a(x0), b)), F(a(x0), b), A(x0)) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCODE_A(z0) -> c(A(encArg(z0))) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_f(z0, z1))) -> c1(A(f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, cons_a(z0))) -> c2(F(encArg(x0), a(encArg(z0))), ENCARG(x0), ENCARG(cons_a(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(x0)) -> c8(A(c_f(a(x0), b)), F(a(x0), b), A(x0)) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(x0)) -> c8(A(c_f(a(x0), b)), F(a(x0), b), A(x0)) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCODE_A_1, ENCODE_F_2, ENCARG_1, A_1 Compound Symbols: c10_1, c_1, c1_2, c2_3, c2_2, c2_1, c7_4, c8_3 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCODE_A(z0) -> c(A(encArg(z0))) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_f(z0, z1))) -> c1(A(f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, cons_a(z0))) -> c2(F(encArg(x0), a(encArg(z0))), ENCARG(x0), ENCARG(cons_a(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCODE_A_1, ENCODE_F_2, ENCARG_1, A_1 Compound Symbols: c10_1, c_1, c1_2, c2_3, c2_2, c2_1, c7_4, c8_3, c8_2 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_A(z0) -> c(A(encArg(z0))) by ENCODE_A(b) -> c(A(b)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_f(z0, z1))) -> c1(A(f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, cons_a(z0))) -> c2(F(encArg(x0), a(encArg(z0))), ENCARG(x0), ENCARG(cons_a(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(b) -> c(A(b)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCODE_F_2, ENCARG_1, A_1, ENCODE_A_1 Compound Symbols: c10_1, c_1, c1_2, c2_3, c2_2, c2_1, c7_4, c8_3, c8_2 ---------------------------------------- (65) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_A(b) -> c(A(b)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_f(z0, z1))) -> c1(A(f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, cons_a(z0))) -> c2(F(encArg(x0), a(encArg(z0))), ENCARG(x0), ENCARG(cons_a(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCODE_F_2, ENCARG_1, A_1, ENCODE_A_1 Compound Symbols: c10_1, c_1, c1_2, c2_3, c2_2, c2_1, c7_4, c8_3, c8_2 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) by ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(b, x1) -> c(F(b, encArg(x1))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_f(z0, z1))) -> c1(A(f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, cons_a(z0))) -> c2(F(encArg(x0), a(encArg(z0))), ENCARG(x0), ENCARG(cons_a(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(b, x1) -> c(F(b, encArg(x1))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c1_2, c2_3, c2_2, c2_1, c7_4, c8_3, c8_2, c_1 ---------------------------------------- (69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_F(b, x1) -> c(F(b, encArg(x1))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_f(z0, z1))) -> c1(A(f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, cons_a(z0))) -> c2(F(encArg(x0), a(encArg(z0))), ENCARG(x0), ENCARG(cons_a(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c1_2, c2_3, c2_2, c2_1, c7_4, c8_3, c8_2, c_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(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b)), ENCARG(cons_a(b))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_a(cons_f(z0, z1))) -> c1(A(f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, cons_a(z0))) -> c2(F(encArg(x0), a(encArg(z0))), ENCARG(x0), ENCARG(cons_a(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b)), ENCARG(cons_a(b))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c1_2, c2_3, c2_2, c2_1, c7_4, c8_3, c8_2, c_1 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_a(cons_f(z0, z1))) -> c1(A(f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, cons_a(z0))) -> c2(F(encArg(x0), a(encArg(z0))), ENCARG(x0), ENCARG(cons_a(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c1_2, c2_3, c2_2, c2_1, c7_4, c8_3, c8_2, c_1, c1_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_a(cons_f(z0, z1))) -> c1(A(f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) by ENCARG(cons_a(cons_f(x0, x1))) -> c1(A(c_f(encArg(x0), encArg(x1))), ENCARG(cons_f(x0, x1))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(x0, cons_a(z0))) -> c2(F(encArg(x0), a(encArg(z0))), ENCARG(x0), ENCARG(cons_a(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(A(c_f(encArg(x0), encArg(x1))), ENCARG(cons_f(x0, x1))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_3, c2_2, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(x0, cons_a(z0))) -> c2(F(encArg(x0), a(encArg(z0))), ENCARG(x0), ENCARG(cons_a(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_3, c2_2, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(x0, cons_a(z0))) -> c2(F(encArg(x0), a(encArg(z0))), ENCARG(x0), ENCARG(cons_a(z0))) by ENCARG(cons_f(x0, cons_a(x1))) -> c2(F(encArg(x0), c_a(encArg(x1))), ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0), ENCARG(cons_a(b))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(b, cons_a(x1))) -> c2(F(b, a(encArg(x1))), ENCARG(b), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(F(encArg(x0), c_a(encArg(x1))), ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0), ENCARG(cons_a(b))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(b, cons_a(x1))) -> c2(F(b, a(encArg(x1))), ENCARG(b), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_3, c2_2, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1 ---------------------------------------- (81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_3, c2_2, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) by ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(F(b, f(encArg(x1), encArg(x2))), ENCARG(b), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(F(b, f(encArg(x1), encArg(x2))), ENCARG(b), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_3, c2_2, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1 ---------------------------------------- (85) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_3, c2_2, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(cons_a(z0), x1)) -> c2(F(a(encArg(z0)), encArg(x1)), ENCARG(cons_a(z0)), ENCARG(x1)) by ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0)), ENCARG(b)) ENCARG(cons_f(cons_a(x0), cons_a(z0))) -> c2(F(a(encArg(x0)), a(encArg(z0))), ENCARG(cons_a(x0)), ENCARG(cons_a(z0))) ENCARG(cons_f(cons_a(x0), cons_f(z0, z1))) -> c2(F(a(encArg(x0)), f(encArg(z0), encArg(z1))), ENCARG(cons_a(x0)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(cons_a(b)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0)), ENCARG(b)) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(cons_a(b)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_3, c2_2, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1 ---------------------------------------- (89) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_3, c2_2, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) by ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1)), ENCARG(b)) ENCARG(cons_f(cons_f(x0, x1), cons_a(z0))) -> c2(F(f(encArg(x0), encArg(x1)), a(encArg(z0))), ENCARG(cons_f(x0, x1)), ENCARG(cons_a(z0))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c2(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1)), ENCARG(b)) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_2, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1, c2_3 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_2, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1, c2_3 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(x0, b)) -> c2(F(encArg(x0), b), ENCARG(x0)) by ENCARG(cons_f(b, b)) -> c2(F(b, b), ENCARG(b)) ENCARG(cons_f(cons_a(z0), b)) -> c2(F(a(encArg(z0)), b), ENCARG(cons_a(z0))) ENCARG(cons_f(cons_f(z0, z1), b)) -> c2(F(f(encArg(z0), encArg(z1)), b), ENCARG(cons_f(z0, z1))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(b, b)) -> c2(F(b, b), ENCARG(b)) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1, c2_3, c2_2 ---------------------------------------- (97) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCARG(cons_f(b, b)) -> c2(F(b, b), ENCARG(b)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1, c2_3, c2_2 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) by A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1, c2_3, c2_2 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) by A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1, c2_3, c2_2 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) by A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(a(c_a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(a(c_a(c_a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0)))), A(c_f(b, c_a(z0)))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0)))), A(c_f(b, c_a(z0)))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0)))), A(c_f(b, c_a(z0)))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1, c2_3, c2_2, c7_1 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1, c2_3, c2_2, c7_1, c7_3 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) by A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(c_a(a(f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(c_a(f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c7_4, c8_3, c8_2, c_1, c1_2, c1_1, c2_3, c2_2, c7_1, c7_3 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) by A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(c_a(a(f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(c_a(f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c8_3, c8_2, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(z0)) -> c8(A(f(c_a(z0), b)), F(a(z0), b), A(z0)) by A(c_a(x0)) -> c8(A(c_f(c_a(x0), b)), F(a(x0), b), A(x0)) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(x0)) -> c8(A(c_f(c_a(x0), b)), F(a(x0), b), A(x0)) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(x0)) -> c8(A(c_f(c_a(x0), b)), F(a(x0), b), A(x0)) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c8_3, c8_2, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2 ---------------------------------------- (113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c8_3, c8_2, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) by A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(c_f(f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(c_f(f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(c_f(f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c8_3, c8_2, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c8_3, c8_2, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) by A(c_a(c_a(x0))) -> c8(A(c_f(f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(c_f(f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(c_f(f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c8_2, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_1 ---------------------------------------- (121) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c8_2, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_1 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) by A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c8_2, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_1 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_A(cons_a(z0)) -> c(A(a(encArg(z0)))) by ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c8_2, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_1 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_A(cons_f(z0, z1)) -> c(A(f(encArg(z0), encArg(z1)))) by ENCODE_A(cons_f(x0, x1)) -> c(A(c_f(encArg(x0), encArg(x1)))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(x0, x1)) -> c(A(c_f(encArg(x0), encArg(x1)))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_F_2, ENCODE_A_1 Compound Symbols: c10_1, c2_1, c8_2, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_1 ---------------------------------------- (129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_A(cons_f(x0, x1)) -> c(A(c_f(encArg(x0), encArg(x1)))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) ENCODE_F(x0, b) -> c(F(encArg(x0), b)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2, A_1, ENCODE_A_1 Compound Symbols: c10_1, c2_1, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_2, c8_1 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(x0, b) -> c(F(encArg(x0), b)) by ENCODE_F(b, b) -> c(F(b, b)) ENCODE_F(cons_a(z0), b) -> c(F(a(encArg(z0)), b)) ENCODE_F(cons_f(z0, z1), b) -> c(F(f(encArg(z0), encArg(z1)), b)) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) ENCODE_F(b, b) -> c(F(b, b)) ENCODE_F(cons_a(z0), b) -> c(F(a(encArg(z0)), b)) ENCODE_F(cons_f(z0, z1), b) -> c(F(f(encArg(z0), encArg(z1)), b)) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2, A_1, ENCODE_A_1 Compound Symbols: c10_1, c2_1, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_2, c8_1 ---------------------------------------- (133) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_F(b, b) -> c(F(b, b)) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) ENCODE_F(cons_a(z0), b) -> c(F(a(encArg(z0)), b)) ENCODE_F(cons_f(z0, z1), b) -> c(F(f(encArg(z0), encArg(z1)), b)) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2, A_1, ENCODE_A_1 Compound Symbols: c10_1, c2_1, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_2, c8_1 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(x0, cons_a(z0)) -> c(F(encArg(x0), a(encArg(z0)))) by ENCODE_F(x0, cons_a(x1)) -> c(F(encArg(x0), c_a(encArg(x1)))) ENCODE_F(x0, cons_a(b)) -> c(F(encArg(x0), a(b))) ENCODE_F(x0, cons_a(cons_a(z0))) -> c(F(encArg(x0), a(a(encArg(z0))))) ENCODE_F(x0, cons_a(cons_f(z0, z1))) -> c(F(encArg(x0), a(f(encArg(z0), encArg(z1))))) ENCODE_F(b, cons_a(x1)) -> c(F(b, a(encArg(x1)))) ENCODE_F(cons_a(z0), cons_a(x1)) -> c(F(a(encArg(z0)), a(encArg(x1)))) ENCODE_F(cons_f(z0, z1), cons_a(x1)) -> c(F(f(encArg(z0), encArg(z1)), a(encArg(x1)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) ENCODE_F(cons_a(z0), b) -> c(F(a(encArg(z0)), b)) ENCODE_F(cons_f(z0, z1), b) -> c(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(x0, cons_a(x1)) -> c(F(encArg(x0), c_a(encArg(x1)))) ENCODE_F(x0, cons_a(b)) -> c(F(encArg(x0), a(b))) ENCODE_F(x0, cons_a(cons_a(z0))) -> c(F(encArg(x0), a(a(encArg(z0))))) ENCODE_F(x0, cons_a(cons_f(z0, z1))) -> c(F(encArg(x0), a(f(encArg(z0), encArg(z1))))) ENCODE_F(b, cons_a(x1)) -> c(F(b, a(encArg(x1)))) ENCODE_F(cons_a(z0), cons_a(x1)) -> c(F(a(encArg(z0)), a(encArg(x1)))) ENCODE_F(cons_f(z0, z1), cons_a(x1)) -> c(F(f(encArg(z0), encArg(z1)), a(encArg(x1)))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2, A_1, ENCODE_A_1 Compound Symbols: c10_1, c2_1, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_2, c8_1 ---------------------------------------- (137) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: ENCODE_F(x0, cons_a(x1)) -> c(F(encArg(x0), c_a(encArg(x1)))) ENCODE_F(b, cons_a(x1)) -> c(F(b, a(encArg(x1)))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) ENCODE_F(cons_a(z0), b) -> c(F(a(encArg(z0)), b)) ENCODE_F(cons_f(z0, z1), b) -> c(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(x0, cons_a(b)) -> c(F(encArg(x0), a(b))) ENCODE_F(x0, cons_a(cons_a(z0))) -> c(F(encArg(x0), a(a(encArg(z0))))) ENCODE_F(x0, cons_a(cons_f(z0, z1))) -> c(F(encArg(x0), a(f(encArg(z0), encArg(z1))))) ENCODE_F(cons_a(z0), cons_a(x1)) -> c(F(a(encArg(z0)), a(encArg(x1)))) ENCODE_F(cons_f(z0, z1), cons_a(x1)) -> c(F(f(encArg(z0), encArg(z1)), a(encArg(x1)))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2, A_1, ENCODE_A_1 Compound Symbols: c10_1, c2_1, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_2, c8_1 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) by ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, b)) -> c(F(encArg(x0), f(encArg(x1), b))) ENCODE_F(x0, cons_f(x1, cons_a(z0))) -> c(F(encArg(x0), f(encArg(x1), a(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(b, x2)) -> c(F(encArg(x0), f(b, encArg(x2)))) ENCODE_F(x0, cons_f(cons_a(z0), x2)) -> c(F(encArg(x0), f(a(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(b, cons_f(x1, x2)) -> c(F(b, f(encArg(x1), encArg(x2)))) ENCODE_F(cons_a(z0), cons_f(x1, x2)) -> c(F(a(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) ENCODE_F(cons_a(z0), b) -> c(F(a(encArg(z0)), b)) ENCODE_F(cons_f(z0, z1), b) -> c(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(x0, cons_a(b)) -> c(F(encArg(x0), a(b))) ENCODE_F(x0, cons_a(cons_a(z0))) -> c(F(encArg(x0), a(a(encArg(z0))))) ENCODE_F(x0, cons_a(cons_f(z0, z1))) -> c(F(encArg(x0), a(f(encArg(z0), encArg(z1))))) ENCODE_F(cons_a(z0), cons_a(x1)) -> c(F(a(encArg(z0)), a(encArg(x1)))) ENCODE_F(cons_f(z0, z1), cons_a(x1)) -> c(F(f(encArg(z0), encArg(z1)), a(encArg(x1)))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, b)) -> c(F(encArg(x0), f(encArg(x1), b))) ENCODE_F(x0, cons_f(x1, cons_a(z0))) -> c(F(encArg(x0), f(encArg(x1), a(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(b, x2)) -> c(F(encArg(x0), f(b, encArg(x2)))) ENCODE_F(x0, cons_f(cons_a(z0), x2)) -> c(F(encArg(x0), f(a(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(b, cons_f(x1, x2)) -> c(F(b, f(encArg(x1), encArg(x2)))) ENCODE_F(cons_a(z0), cons_f(x1, x2)) -> c(F(a(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2, A_1, ENCODE_A_1 Compound Symbols: c10_1, c2_1, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_2, c8_1 ---------------------------------------- (141) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: ENCODE_F(b, cons_f(x1, x2)) -> c(F(b, f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) ENCODE_F(cons_a(z0), b) -> c(F(a(encArg(z0)), b)) ENCODE_F(cons_f(z0, z1), b) -> c(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(x0, cons_a(b)) -> c(F(encArg(x0), a(b))) ENCODE_F(x0, cons_a(cons_a(z0))) -> c(F(encArg(x0), a(a(encArg(z0))))) ENCODE_F(x0, cons_a(cons_f(z0, z1))) -> c(F(encArg(x0), a(f(encArg(z0), encArg(z1))))) ENCODE_F(cons_a(z0), cons_a(x1)) -> c(F(a(encArg(z0)), a(encArg(x1)))) ENCODE_F(cons_f(z0, z1), cons_a(x1)) -> c(F(f(encArg(z0), encArg(z1)), a(encArg(x1)))) ENCODE_F(x0, cons_f(x1, b)) -> c(F(encArg(x0), f(encArg(x1), b))) ENCODE_F(x0, cons_f(x1, cons_a(z0))) -> c(F(encArg(x0), f(encArg(x1), a(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(b, x2)) -> c(F(encArg(x0), f(b, encArg(x2)))) ENCODE_F(x0, cons_f(cons_a(z0), x2)) -> c(F(encArg(x0), f(a(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_a(z0), cons_f(x1, x2)) -> c(F(a(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2, A_1, ENCODE_A_1 Compound Symbols: c10_1, c2_1, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_2, c8_1 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(cons_a(z0), x1) -> c(F(a(encArg(z0)), encArg(x1))) by ENCODE_F(cons_a(x0), b) -> c(F(a(encArg(x0)), b)) ENCODE_F(cons_a(x0), cons_a(z0)) -> c(F(a(encArg(x0)), a(encArg(z0)))) ENCODE_F(cons_a(x0), cons_f(z0, z1)) -> c(F(a(encArg(x0)), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_a(x0), x1) -> c(F(c_a(encArg(x0)), encArg(x1))) ENCODE_F(cons_a(b), x1) -> c(F(a(b), encArg(x1))) ENCODE_F(cons_a(cons_a(z0)), x1) -> c(F(a(a(encArg(z0))), encArg(x1))) ENCODE_F(cons_a(cons_f(z0, z1)), x1) -> c(F(a(f(encArg(z0), encArg(z1))), encArg(x1))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) ENCODE_F(cons_a(z0), b) -> c(F(a(encArg(z0)), b)) ENCODE_F(cons_f(z0, z1), b) -> c(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(x0, cons_a(b)) -> c(F(encArg(x0), a(b))) ENCODE_F(x0, cons_a(cons_a(z0))) -> c(F(encArg(x0), a(a(encArg(z0))))) ENCODE_F(x0, cons_a(cons_f(z0, z1))) -> c(F(encArg(x0), a(f(encArg(z0), encArg(z1))))) ENCODE_F(cons_a(z0), cons_a(x1)) -> c(F(a(encArg(z0)), a(encArg(x1)))) ENCODE_F(cons_f(z0, z1), cons_a(x1)) -> c(F(f(encArg(z0), encArg(z1)), a(encArg(x1)))) ENCODE_F(x0, cons_f(x1, b)) -> c(F(encArg(x0), f(encArg(x1), b))) ENCODE_F(x0, cons_f(x1, cons_a(z0))) -> c(F(encArg(x0), f(encArg(x1), a(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(b, x2)) -> c(F(encArg(x0), f(b, encArg(x2)))) ENCODE_F(x0, cons_f(cons_a(z0), x2)) -> c(F(encArg(x0), f(a(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_a(z0), cons_f(x1, x2)) -> c(F(a(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_a(x0), x1) -> c(F(c_a(encArg(x0)), encArg(x1))) ENCODE_F(cons_a(b), x1) -> c(F(a(b), encArg(x1))) ENCODE_F(cons_a(cons_a(z0)), x1) -> c(F(a(a(encArg(z0))), encArg(x1))) ENCODE_F(cons_a(cons_f(z0, z1)), x1) -> c(F(a(f(encArg(z0), encArg(z1))), encArg(x1))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2, A_1, ENCODE_A_1 Compound Symbols: c10_1, c2_1, c_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_2, c8_1 ---------------------------------------- (145) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) by ENCODE_F(cons_f(x0, x1), b) -> c(F(f(encArg(x0), encArg(x1)), b)) ENCODE_F(cons_f(x0, x1), cons_a(z0)) -> c(F(f(encArg(x0), encArg(x1)), a(encArg(z0)))) ENCODE_F(cons_f(x0, x1), cons_f(z0, z1)) -> c(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, b), x2) -> c(F(f(encArg(x0), b), encArg(x2))) ENCODE_F(cons_f(x0, cons_a(z0)), x2) -> c(F(f(encArg(x0), a(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(b, x1), x2) -> c(F(f(b, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_a(z0), x1), x2) -> c(F(f(a(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) ENCODE_F(cons_a(z0), b) -> c(F(a(encArg(z0)), b)) ENCODE_F(cons_f(z0, z1), b) -> c(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(x0, cons_a(b)) -> c(F(encArg(x0), a(b))) ENCODE_F(x0, cons_a(cons_a(z0))) -> c(F(encArg(x0), a(a(encArg(z0))))) ENCODE_F(x0, cons_a(cons_f(z0, z1))) -> c(F(encArg(x0), a(f(encArg(z0), encArg(z1))))) ENCODE_F(cons_a(z0), cons_a(x1)) -> c(F(a(encArg(z0)), a(encArg(x1)))) ENCODE_F(cons_f(z0, z1), cons_a(x1)) -> c(F(f(encArg(z0), encArg(z1)), a(encArg(x1)))) ENCODE_F(x0, cons_f(x1, b)) -> c(F(encArg(x0), f(encArg(x1), b))) ENCODE_F(x0, cons_f(x1, cons_a(z0))) -> c(F(encArg(x0), f(encArg(x1), a(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(b, x2)) -> c(F(encArg(x0), f(b, encArg(x2)))) ENCODE_F(x0, cons_f(cons_a(z0), x2)) -> c(F(encArg(x0), f(a(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_a(z0), cons_f(x1, x2)) -> c(F(a(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_a(x0), x1) -> c(F(c_a(encArg(x0)), encArg(x1))) ENCODE_F(cons_a(b), x1) -> c(F(a(b), encArg(x1))) ENCODE_F(cons_a(cons_a(z0)), x1) -> c(F(a(a(encArg(z0))), encArg(x1))) ENCODE_F(cons_a(cons_f(z0, z1)), x1) -> c(F(a(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, b), x2) -> c(F(f(encArg(x0), b), encArg(x2))) ENCODE_F(cons_f(x0, cons_a(z0)), x2) -> c(F(f(encArg(x0), a(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(b, x1), x2) -> c(F(f(b, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_a(z0), x1), x2) -> c(F(f(a(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_2, c8_1, c_1 ---------------------------------------- (147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) ENCODE_F(cons_a(z0), b) -> c(F(a(encArg(z0)), b)) ENCODE_F(cons_f(z0, z1), b) -> c(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(x0, cons_a(b)) -> c(F(encArg(x0), a(b))) ENCODE_F(x0, cons_a(cons_a(z0))) -> c(F(encArg(x0), a(a(encArg(z0))))) ENCODE_F(x0, cons_a(cons_f(z0, z1))) -> c(F(encArg(x0), a(f(encArg(z0), encArg(z1))))) ENCODE_F(cons_a(z0), cons_a(x1)) -> c(F(a(encArg(z0)), a(encArg(x1)))) ENCODE_F(cons_f(z0, z1), cons_a(x1)) -> c(F(f(encArg(z0), encArg(z1)), a(encArg(x1)))) ENCODE_F(x0, cons_f(x1, b)) -> c(F(encArg(x0), f(encArg(x1), b))) ENCODE_F(x0, cons_f(x1, cons_a(z0))) -> c(F(encArg(x0), f(encArg(x1), a(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(b, x2)) -> c(F(encArg(x0), f(b, encArg(x2)))) ENCODE_F(x0, cons_f(cons_a(z0), x2)) -> c(F(encArg(x0), f(a(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_a(z0), cons_f(x1, x2)) -> c(F(a(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_a(x0), x1) -> c(F(c_a(encArg(x0)), encArg(x1))) ENCODE_F(cons_a(b), x1) -> c(F(a(b), encArg(x1))) ENCODE_F(cons_a(cons_a(z0)), x1) -> c(F(a(a(encArg(z0))), encArg(x1))) ENCODE_F(cons_a(cons_f(z0, z1)), x1) -> c(F(a(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, b), x2) -> c(F(f(encArg(x0), b), encArg(x2))) ENCODE_F(cons_f(x0, cons_a(z0)), x2) -> c(F(f(encArg(x0), a(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(b, x1), x2) -> c(F(f(b, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_a(z0), x1), x2) -> c(F(f(a(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_2, c8_1, c_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ENCARG(cons_a(cons_a(b))) -> c1(A(a(b))) by ENCARG(cons_a(cons_a(b))) -> c1(A(c_a(b))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) ENCODE_F(cons_a(z0), b) -> c(F(a(encArg(z0)), b)) ENCODE_F(cons_f(z0, z1), b) -> c(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(x0, cons_a(b)) -> c(F(encArg(x0), a(b))) ENCODE_F(x0, cons_a(cons_a(z0))) -> c(F(encArg(x0), a(a(encArg(z0))))) ENCODE_F(x0, cons_a(cons_f(z0, z1))) -> c(F(encArg(x0), a(f(encArg(z0), encArg(z1))))) ENCODE_F(cons_a(z0), cons_a(x1)) -> c(F(a(encArg(z0)), a(encArg(x1)))) ENCODE_F(cons_f(z0, z1), cons_a(x1)) -> c(F(f(encArg(z0), encArg(z1)), a(encArg(x1)))) ENCODE_F(x0, cons_f(x1, b)) -> c(F(encArg(x0), f(encArg(x1), b))) ENCODE_F(x0, cons_f(x1, cons_a(z0))) -> c(F(encArg(x0), f(encArg(x1), a(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(b, x2)) -> c(F(encArg(x0), f(b, encArg(x2)))) ENCODE_F(x0, cons_f(cons_a(z0), x2)) -> c(F(encArg(x0), f(a(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_a(z0), cons_f(x1, x2)) -> c(F(a(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_a(x0), x1) -> c(F(c_a(encArg(x0)), encArg(x1))) ENCODE_F(cons_a(b), x1) -> c(F(a(b), encArg(x1))) ENCODE_F(cons_a(cons_a(z0)), x1) -> c(F(a(a(encArg(z0))), encArg(x1))) ENCODE_F(cons_a(cons_f(z0, z1)), x1) -> c(F(a(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, b), x2) -> c(F(f(encArg(x0), b), encArg(x2))) ENCODE_F(cons_f(x0, cons_a(z0)), x2) -> c(F(f(encArg(x0), a(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(b, x1), x2) -> c(F(f(b, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_a(z0), x1), x2) -> c(F(f(a(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_a(cons_a(b))) -> c1(A(c_a(b))) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_2, c8_1, c_1 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ENCARG(cons_f(x0, cons_a(b))) -> c2(F(encArg(x0), a(b)), ENCARG(x0)) by ENCARG(cons_f(z0, cons_a(b))) -> c2(F(encArg(z0), c_a(b)), ENCARG(z0)) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) ENCODE_F(cons_a(z0), b) -> c(F(a(encArg(z0)), b)) ENCODE_F(cons_f(z0, z1), b) -> c(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(x0, cons_a(b)) -> c(F(encArg(x0), a(b))) ENCODE_F(x0, cons_a(cons_a(z0))) -> c(F(encArg(x0), a(a(encArg(z0))))) ENCODE_F(x0, cons_a(cons_f(z0, z1))) -> c(F(encArg(x0), a(f(encArg(z0), encArg(z1))))) ENCODE_F(cons_a(z0), cons_a(x1)) -> c(F(a(encArg(z0)), a(encArg(x1)))) ENCODE_F(cons_f(z0, z1), cons_a(x1)) -> c(F(f(encArg(z0), encArg(z1)), a(encArg(x1)))) ENCODE_F(x0, cons_f(x1, b)) -> c(F(encArg(x0), f(encArg(x1), b))) ENCODE_F(x0, cons_f(x1, cons_a(z0))) -> c(F(encArg(x0), f(encArg(x1), a(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(b, x2)) -> c(F(encArg(x0), f(b, encArg(x2)))) ENCODE_F(x0, cons_f(cons_a(z0), x2)) -> c(F(encArg(x0), f(a(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_a(z0), cons_f(x1, x2)) -> c(F(a(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_a(x0), x1) -> c(F(c_a(encArg(x0)), encArg(x1))) ENCODE_F(cons_a(b), x1) -> c(F(a(b), encArg(x1))) ENCODE_F(cons_a(cons_a(z0)), x1) -> c(F(a(a(encArg(z0))), encArg(x1))) ENCODE_F(cons_a(cons_f(z0, z1)), x1) -> c(F(a(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, b), x2) -> c(F(f(encArg(x0), b), encArg(x2))) ENCODE_F(cons_f(x0, cons_a(z0)), x2) -> c(F(f(encArg(x0), a(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(b, x1), x2) -> c(F(f(b, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_a(z0), x1), x2) -> c(F(f(a(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_a(cons_a(b))) -> c1(A(c_a(b))) ENCARG(cons_f(z0, cons_a(b))) -> c2(F(encArg(z0), c_a(b)), ENCARG(z0)) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_2, c8_1, c_1 ---------------------------------------- (153) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) ENCODE_F(cons_a(z0), b) -> c(F(a(encArg(z0)), b)) ENCODE_F(cons_f(z0, z1), b) -> c(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(x0, cons_a(b)) -> c(F(encArg(x0), a(b))) ENCODE_F(x0, cons_a(cons_a(z0))) -> c(F(encArg(x0), a(a(encArg(z0))))) ENCODE_F(x0, cons_a(cons_f(z0, z1))) -> c(F(encArg(x0), a(f(encArg(z0), encArg(z1))))) ENCODE_F(cons_a(z0), cons_a(x1)) -> c(F(a(encArg(z0)), a(encArg(x1)))) ENCODE_F(cons_f(z0, z1), cons_a(x1)) -> c(F(f(encArg(z0), encArg(z1)), a(encArg(x1)))) ENCODE_F(x0, cons_f(x1, b)) -> c(F(encArg(x0), f(encArg(x1), b))) ENCODE_F(x0, cons_f(x1, cons_a(z0))) -> c(F(encArg(x0), f(encArg(x1), a(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(b, x2)) -> c(F(encArg(x0), f(b, encArg(x2)))) ENCODE_F(x0, cons_f(cons_a(z0), x2)) -> c(F(encArg(x0), f(a(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_a(z0), cons_f(x1, x2)) -> c(F(a(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_a(x0), x1) -> c(F(c_a(encArg(x0)), encArg(x1))) ENCODE_F(cons_a(b), x1) -> c(F(a(b), encArg(x1))) ENCODE_F(cons_a(cons_a(z0)), x1) -> c(F(a(a(encArg(z0))), encArg(x1))) ENCODE_F(cons_a(cons_f(z0, z1)), x1) -> c(F(a(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, b), x2) -> c(F(f(encArg(x0), b), encArg(x2))) ENCODE_F(cons_f(x0, cons_a(z0)), x2) -> c(F(f(encArg(x0), a(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(b, x1), x2) -> c(F(f(b, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_a(z0), x1), x2) -> c(F(f(a(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_a(cons_a(b))) -> c1(A(c_a(b))) ENCARG(cons_f(z0, cons_a(b))) -> c2(ENCARG(z0)) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_2, c8_1, c_1 ---------------------------------------- (155) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ENCARG(cons_f(cons_a(b), x1)) -> c2(F(a(b), encArg(x1)), ENCARG(x1)) by ENCARG(cons_f(cons_a(b), z0)) -> c2(F(c_a(b), encArg(z0)), ENCARG(z0)) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: encArg(b) -> b encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) a(z0) -> c_a(z0) a(c_a(c_f(b, c_a(z0)))) -> f(a(a(a(z0))), b) a(c_a(z0)) -> f(b, a(f(a(z0), b))) f(z0, z1) -> c_f(z0, z1) f(c_a(z0), b) -> f(b, a(z0)) Tuples: F(c_a(z0), b) -> c10(A(z0)) ENCARG(cons_f(b, x1)) -> c2(ENCARG(x1)) ENCARG(cons_a(cons_a(x0))) -> c1(A(c_a(encArg(x0))), ENCARG(cons_a(x0))) 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_f(z0, z1)))) -> c1(A(a(f(encArg(z0), encArg(z1)))), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_a(cons_f(x0, b))) -> c1(A(f(encArg(x0), b)), ENCARG(cons_f(x0, b))) ENCARG(cons_a(cons_f(x0, cons_a(z0)))) -> c1(A(f(encArg(x0), a(encArg(z0)))), ENCARG(cons_f(x0, cons_a(z0)))) ENCARG(cons_a(cons_f(x0, cons_f(z0, z1)))) -> c1(A(f(encArg(x0), f(encArg(z0), encArg(z1)))), ENCARG(cons_f(x0, cons_f(z0, z1)))) ENCARG(cons_a(cons_f(b, x1))) -> c1(A(f(b, encArg(x1))), ENCARG(cons_f(b, x1))) ENCARG(cons_a(cons_f(cons_a(z0), x1))) -> c1(A(f(a(encArg(z0)), encArg(x1))), ENCARG(cons_f(cons_a(z0), x1))) ENCARG(cons_a(cons_f(cons_f(z0, z1), x1))) -> c1(A(f(f(encArg(z0), encArg(z1)), encArg(x1))), ENCARG(cons_f(cons_f(z0, z1), x1))) ENCARG(cons_a(cons_f(x0, x1))) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(x0, cons_a(cons_a(z0)))) -> c2(F(encArg(x0), a(a(encArg(z0)))), ENCARG(x0), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_f(x0, cons_a(cons_f(z0, z1)))) -> c2(F(encArg(x0), a(f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_a(cons_f(z0, z1)))) ENCARG(cons_f(cons_a(z0), cons_a(x1))) -> c2(F(a(encArg(z0)), a(encArg(x1))), ENCARG(cons_a(z0)), ENCARG(cons_a(x1))) ENCARG(cons_f(cons_f(z0, z1), cons_a(x1))) -> c2(F(f(encArg(z0), encArg(z1)), a(encArg(x1))), ENCARG(cons_f(z0, z1)), ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_a(x1))) -> c2(ENCARG(x0), ENCARG(cons_a(x1))) ENCARG(cons_f(b, cons_a(x1))) -> c2(ENCARG(cons_a(x1))) ENCARG(cons_f(x0, cons_f(x1, b))) -> c2(F(encArg(x0), f(encArg(x1), b)), ENCARG(x0), ENCARG(cons_f(x1, b))) ENCARG(cons_f(x0, cons_f(x1, cons_a(z0)))) -> c2(F(encArg(x0), f(encArg(x1), a(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, cons_a(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(b, x2))) -> c2(F(encArg(x0), f(b, encArg(x2))), ENCARG(x0), ENCARG(cons_f(b, x2))) ENCARG(cons_f(x0, cons_f(cons_a(z0), x2))) -> c2(F(encArg(x0), f(a(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_a(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_a(z0), cons_f(x1, x2))) -> c2(F(a(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(cons_a(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(b, cons_f(x1, x2))) -> c2(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_a(x0), x1)) -> c2(F(c_a(encArg(x0)), encArg(x1)), ENCARG(cons_a(x0)), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_a(z0)), x1)) -> c2(F(a(a(encArg(z0))), encArg(x1)), ENCARG(cons_a(cons_a(z0))), ENCARG(x1)) ENCARG(cons_f(cons_a(cons_f(z0, z1)), x1)) -> c2(F(a(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(cons_a(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(cons_a(x0), b)) -> c2(F(a(encArg(x0)), b), ENCARG(cons_a(x0))) ENCARG(cons_f(cons_f(x0, b), x2)) -> c2(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_a(z0)), x2)) -> c2(F(f(encArg(x0), a(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, cons_a(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c2(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_a(z0), x1), x2)) -> c2(F(f(a(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_a(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c2(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(x2)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(x0)) -> c8(F(a(x0), b), A(x0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) ENCODE_A(cons_a(x0)) -> c(A(c_a(encArg(x0)))) ENCODE_A(cons_a(b)) -> c(A(a(b))) ENCODE_A(cons_a(cons_a(z0))) -> c(A(a(a(encArg(z0))))) ENCODE_A(cons_a(cons_f(z0, z1))) -> c(A(a(f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(x0, b)) -> c(A(f(encArg(x0), b))) ENCODE_A(cons_f(x0, cons_a(z0))) -> c(A(f(encArg(x0), a(encArg(z0))))) ENCODE_A(cons_f(x0, cons_f(z0, z1))) -> c(A(f(encArg(x0), f(encArg(z0), encArg(z1))))) ENCODE_A(cons_f(b, x1)) -> c(A(f(b, encArg(x1)))) ENCODE_A(cons_f(cons_a(z0), x1)) -> c(A(f(a(encArg(z0)), encArg(x1)))) ENCODE_A(cons_f(cons_f(z0, z1), x1)) -> c(A(f(f(encArg(z0), encArg(z1)), encArg(x1)))) ENCODE_F(cons_a(z0), b) -> c(F(a(encArg(z0)), b)) ENCODE_F(cons_f(z0, z1), b) -> c(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(x0, cons_a(b)) -> c(F(encArg(x0), a(b))) ENCODE_F(x0, cons_a(cons_a(z0))) -> c(F(encArg(x0), a(a(encArg(z0))))) ENCODE_F(x0, cons_a(cons_f(z0, z1))) -> c(F(encArg(x0), a(f(encArg(z0), encArg(z1))))) ENCODE_F(cons_a(z0), cons_a(x1)) -> c(F(a(encArg(z0)), a(encArg(x1)))) ENCODE_F(cons_f(z0, z1), cons_a(x1)) -> c(F(f(encArg(z0), encArg(z1)), a(encArg(x1)))) ENCODE_F(x0, cons_f(x1, b)) -> c(F(encArg(x0), f(encArg(x1), b))) ENCODE_F(x0, cons_f(x1, cons_a(z0))) -> c(F(encArg(x0), f(encArg(x1), a(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(b, x2)) -> c(F(encArg(x0), f(b, encArg(x2)))) ENCODE_F(x0, cons_f(cons_a(z0), x2)) -> c(F(encArg(x0), f(a(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_a(z0), cons_f(x1, x2)) -> c(F(a(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_a(x0), x1) -> c(F(c_a(encArg(x0)), encArg(x1))) ENCODE_F(cons_a(b), x1) -> c(F(a(b), encArg(x1))) ENCODE_F(cons_a(cons_a(z0)), x1) -> c(F(a(a(encArg(z0))), encArg(x1))) ENCODE_F(cons_a(cons_f(z0, z1)), x1) -> c(F(a(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, b), x2) -> c(F(f(encArg(x0), b), encArg(x2))) ENCODE_F(cons_f(x0, cons_a(z0)), x2) -> c(F(f(encArg(x0), a(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(b, x1), x2) -> c(F(f(b, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_a(z0), x1), x2) -> c(F(f(a(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_a(cons_a(b))) -> c1(A(c_a(b))) ENCARG(cons_f(z0, cons_a(b))) -> c2(ENCARG(z0)) ENCARG(cons_f(cons_a(b), z0)) -> c2(F(c_a(b), encArg(z0)), ENCARG(z0)) S tuples: F(c_a(z0), b) -> c10(A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(F(c_a(c_a(a(x0))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(c_a(a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(c_a(a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(c_a(a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(x0)))) -> c7(F(f(b, a(f(a(a(x0)), b))), b), A(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(c_a(c_a(z0))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(c_a(f(a(a(a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(c_a(f(b, a(f(a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(z0)))) -> c7(F(a(f(b, a(f(a(z0), b)))), b), A(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_f(b, c_a(x0)))) -> c7(A(a(a(x0)))) A(c_a(c_f(b, c_a(c_f(b, c_a(z0)))))) -> c7(F(a(f(a(a(a(z0))), b)), b), A(a(a(c_f(b, c_a(z0))))), A(a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(c_f(a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(c_a(a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(F(a(a(f(a(c_a(a(x0))), b))), b), A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))) -> c7(F(a(a(f(a(a(c_a(z0))), b))), b), A(a(a(c_a(c_f(b, c_a(z0)))))), A(a(c_a(c_f(b, c_a(z0))))), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))))) -> c7(F(a(a(f(a(a(f(a(a(a(z0))), b))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))))), A(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(c_a(z0)))))))) -> c7(F(a(a(f(a(a(f(b, a(f(a(z0), b))))), b))), b), A(a(a(c_a(c_f(b, c_a(c_a(z0))))))), A(a(c_a(c_f(b, c_a(c_a(z0)))))), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(x0))))))) -> c7(A(a(a(c_a(c_f(b, c_a(x0)))))), A(a(c_a(c_f(b, c_a(x0))))), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(c_f(b, a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, c_a(f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(F(a(a(f(b, a(c_f(a(x0), b))))), b), A(a(a(c_a(x0)))), A(a(c_a(x0))), A(c_a(x0))) A(c_a(c_f(b, c_a(c_a(z0))))) -> c7(F(a(a(f(b, a(f(c_a(z0), b))))), b), A(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_f(b, c_a(c_a(c_a(c_f(b, c_a(z0)))))))) -> c7(F(a(a(f(b, a(f(f(a(a(a(z0))), b), b))))), b), A(a(a(c_a(c_a(c_f(b, c_a(z0))))))), A(a(c_a(c_a(c_f(b, c_a(z0)))))), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_f(b, c_a(c_a(c_a(z0)))))) -> c7(F(a(a(f(b, a(f(f(b, a(f(a(z0), b))), b))))), b), A(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_f(b, c_a(c_a(x0))))) -> c7(A(a(a(c_a(x0)))), A(a(c_a(x0)))) A(c_a(z0)) -> c8(A(f(b, a(z0))), F(a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(c_f(a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(c_a(a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(A(f(f(a(c_a(a(x0))), b), b)), F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(A(f(f(a(a(c_a(z0))), b), b)), F(a(c_a(c_f(b, c_a(z0)))), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) -> c8(A(f(f(a(a(f(a(a(a(z0))), b))), b), b)), F(a(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0))))))), b), A(c_a(c_f(b, c_a(c_a(c_f(b, c_a(z0)))))))) A(c_a(c_a(c_f(b, c_a(c_a(z0)))))) -> c8(A(f(f(a(a(f(b, a(f(a(z0), b))))), b), b)), F(a(c_a(c_f(b, c_a(c_a(z0))))), b), A(c_a(c_f(b, c_a(c_a(z0)))))) A(c_a(c_a(c_f(b, c_a(x0))))) -> c8(F(a(c_a(c_f(b, c_a(x0)))), b), A(c_a(c_f(b, c_a(x0))))) A(c_a(c_a(x0))) -> c8(A(f(c_f(b, a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, c_a(f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(x0))) -> c8(A(f(f(b, a(c_f(a(x0), b))), b)), F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(c_a(z0))) -> c8(A(f(f(b, a(f(c_a(z0), b))), b)), F(a(c_a(z0)), b), A(c_a(z0))) A(c_a(c_a(c_a(c_f(b, c_a(z0)))))) -> c8(A(f(f(b, a(f(f(a(a(a(z0))), b), b))), b)), F(a(c_a(c_a(c_f(b, c_a(z0))))), b), A(c_a(c_a(c_f(b, c_a(z0)))))) A(c_a(c_a(c_a(z0)))) -> c8(A(f(f(b, a(f(f(b, a(f(a(z0), b))), b))), b)), F(a(c_a(c_a(z0))), b), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b)) A(c_a(c_a(x0))) -> c8(F(a(c_a(x0)), b), A(c_a(x0))) A(c_a(z0)) -> c8(F(c_a(z0), b), A(z0)) A(c_a(c_a(c_f(b, c_a(z0))))) -> c8(F(f(a(a(a(z0))), b), b), A(c_a(c_f(b, c_a(z0))))) A(c_a(c_a(z0))) -> c8(F(f(b, a(f(a(z0), b))), b), A(c_a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, A_1, ENCODE_A_1, ENCODE_F_2 Compound Symbols: c10_1, c2_1, c1_2, c1_1, c2_3, c2_2, c7_4, c7_1, c7_3, c7_2, c8_3, c8_2, c8_1, c_1