/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 40 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), 467 ms] (14) BOUNDS(1, INF) (15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxTRS (17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (18) CpxRelTRS (19) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxWeightedTrs (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxTypedWeightedTrs (23) CompletionProof [UPPER BOUND(ID), 0 ms] (24) CpxTypedWeightedCompleteTrs (25) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (40) CdtProblem (41) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRhsSimplificationProcessorProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: c(b(a(a(x1)))) -> a(a(b(c(x1)))) b(a(a(a(x1)))) -> a(a(a(b(x1)))) a(b(c(x1))) -> c(b(a(x1))) c(c(b(b(x1)))) -> b(b(c(c(x1)))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: c(b(a(a(x1)))) -> a(a(b(c(x1)))) b(a(a(a(x1)))) -> a(a(a(b(x1)))) a(b(c(x1))) -> c(b(a(x1))) c(c(b(b(x1)))) -> b(b(c(c(x1)))) The (relative) TRS S consists of the following rules: encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: c(b(a(a(x1)))) -> a(a(b(c(x1)))) b(a(a(a(x1)))) -> a(a(a(b(x1)))) a(b(c(x1))) -> c(b(a(x1))) c(c(b(b(x1)))) -> b(b(c(c(x1)))) The (relative) TRS S consists of the following rules: encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (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 (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: c(b(a(a(x1)))) -> a(a(b(c(x1)))) b(a(a(a(x1)))) -> a(a(a(b(x1)))) a(b(c(x1))) -> c(b(a(x1))) c(c(b(b(x1)))) -> b(b(c(c(x1)))) The (relative) TRS S consists of the following rules: encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: c(b(a(a(x1)))) -> a(a(b(c(x1)))) b(a(a(a(x1)))) -> a(a(a(b(x1)))) a(b(c(x1))) -> c(b(a(x1))) c(c(b(b(x1)))) -> b(b(c(c(x1)))) The (relative) TRS S consists of the following rules: encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Innermost TRS: Rules: c(b(a(a(x1)))) -> a(a(b(c(x1)))) b(a(a(a(x1)))) -> a(a(a(b(x1)))) a(b(c(x1))) -> c(b(a(x1))) c(c(b(b(x1)))) -> b(b(c(c(x1)))) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Types: c :: a:b:c:encArg:encode_c:encode_b:encode_a -> a:b:c:encArg:encode_c:encode_b:encode_a b :: a:b:c:encArg:encode_c:encode_b:encode_a -> a:b:c:encArg:encode_c:encode_b:encode_a a :: a:b:c:encArg:encode_c:encode_b:encode_a -> a:b:c:encArg:encode_c:encode_b:encode_a encArg :: cons_c:cons_b:cons_a -> a:b:c:encArg:encode_c:encode_b:encode_a cons_c :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a cons_b :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a cons_a :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a encode_c :: cons_c:cons_b:cons_a -> a:b:c:encArg:encode_c:encode_b:encode_a encode_b :: cons_c:cons_b:cons_a -> a:b:c:encArg:encode_c:encode_b:encode_a encode_a :: cons_c:cons_b:cons_a -> a:b:c:encArg:encode_c:encode_b:encode_a hole_a:b:c:encArg:encode_c:encode_b:encode_a1_0 :: a:b:c:encArg:encode_c:encode_b:encode_a hole_cons_c:cons_b:cons_a2_0 :: cons_c:cons_b:cons_a gen_cons_c:cons_b:cons_a3_0 :: Nat -> cons_c:cons_b:cons_a ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: c, a, b, encArg They will be analysed ascendingly in the following order: c = a c = b c < encArg a = b a < encArg b < encArg ---------------------------------------- (12) Obligation: Innermost TRS: Rules: c(b(a(a(x1)))) -> a(a(b(c(x1)))) b(a(a(a(x1)))) -> a(a(a(b(x1)))) a(b(c(x1))) -> c(b(a(x1))) c(c(b(b(x1)))) -> b(b(c(c(x1)))) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Types: c :: a:b:c:encArg:encode_c:encode_b:encode_a -> a:b:c:encArg:encode_c:encode_b:encode_a b :: a:b:c:encArg:encode_c:encode_b:encode_a -> a:b:c:encArg:encode_c:encode_b:encode_a a :: a:b:c:encArg:encode_c:encode_b:encode_a -> a:b:c:encArg:encode_c:encode_b:encode_a encArg :: cons_c:cons_b:cons_a -> a:b:c:encArg:encode_c:encode_b:encode_a cons_c :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a cons_b :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a cons_a :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a encode_c :: cons_c:cons_b:cons_a -> a:b:c:encArg:encode_c:encode_b:encode_a encode_b :: cons_c:cons_b:cons_a -> a:b:c:encArg:encode_c:encode_b:encode_a encode_a :: cons_c:cons_b:cons_a -> a:b:c:encArg:encode_c:encode_b:encode_a hole_a:b:c:encArg:encode_c:encode_b:encode_a1_0 :: a:b:c:encArg:encode_c:encode_b:encode_a hole_cons_c:cons_b:cons_a2_0 :: cons_c:cons_b:cons_a gen_cons_c:cons_b:cons_a3_0 :: Nat -> cons_c:cons_b:cons_a Generator Equations: gen_cons_c:cons_b:cons_a3_0(0) <=> hole_cons_c:cons_b:cons_a2_0 gen_cons_c:cons_b:cons_a3_0(+(x, 1)) <=> cons_c(gen_cons_c:cons_b:cons_a3_0(x)) The following defined symbols remain to be analysed: a, c, b, encArg They will be analysed ascendingly in the following order: c = a c = b c < encArg a = b a < encArg b < encArg ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_cons_c:cons_b:cons_a3_0(+(1, n16_0))) -> *4_0, rt in Omega(0) Induction Base: encArg(gen_cons_c:cons_b:cons_a3_0(+(1, 0))) Induction Step: encArg(gen_cons_c:cons_b:cons_a3_0(+(1, +(n16_0, 1)))) ->_R^Omega(0) c(encArg(gen_cons_c:cons_b:cons_a3_0(+(1, n16_0)))) ->_IH c(*4_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 (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: c(b(a(a(x1)))) -> a(a(b(c(x1)))) b(a(a(a(x1)))) -> a(a(a(b(x1)))) a(b(c(x1))) -> c(b(a(x1))) c(c(b(b(x1)))) -> b(b(c(c(x1)))) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (17) NonCtorToCtorProof (UPPER BOUND(ID)) transformed non-ctor to ctor-system ---------------------------------------- (18) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(c_b(c_c(x1))) -> c(b(a(x1))) c(c_b(c_a(c_a(x1)))) -> a(a(b(c(x1)))) b(c_a(c_a(c_a(x1)))) -> a(a(a(b(x1)))) c(c_c(c_b(c_b(x1)))) -> b(b(c(c(x1)))) The (relative) TRS S consists of the following rules: encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) a(x0) -> c_a(x0) b(x0) -> c_b(x0) c(x0) -> c_c(x0) Rewrite Strategy: INNERMOST ---------------------------------------- (19) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (20) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(c_b(c_c(x1))) -> c(b(a(x1))) [1] c(c_b(c_a(c_a(x1)))) -> a(a(b(c(x1)))) [1] b(c_a(c_a(c_a(x1)))) -> a(a(a(b(x1)))) [1] c(c_c(c_b(c_b(x1)))) -> b(b(c(c(x1)))) [1] encArg(cons_c(x_1)) -> c(encArg(x_1)) [0] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encode_c(x_1) -> c(encArg(x_1)) [0] encode_b(x_1) -> b(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] a(x0) -> c_a(x0) [0] b(x0) -> c_b(x0) [0] c(x0) -> c_c(x0) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (21) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (22) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(c_b(c_c(x1))) -> c(b(a(x1))) [1] c(c_b(c_a(c_a(x1)))) -> a(a(b(c(x1)))) [1] b(c_a(c_a(c_a(x1)))) -> a(a(a(b(x1)))) [1] c(c_c(c_b(c_b(x1)))) -> b(b(c(c(x1)))) [1] encArg(cons_c(x_1)) -> c(encArg(x_1)) [0] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encode_c(x_1) -> c(encArg(x_1)) [0] encode_b(x_1) -> b(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] a(x0) -> c_a(x0) [0] b(x0) -> c_b(x0) [0] c(x0) -> c_c(x0) [0] The TRS has the following type information: a :: c_c:c_b:c_a -> c_c:c_b:c_a c_b :: c_c:c_b:c_a -> c_c:c_b:c_a c_c :: c_c:c_b:c_a -> c_c:c_b:c_a c :: c_c:c_b:c_a -> c_c:c_b:c_a b :: c_c:c_b:c_a -> c_c:c_b:c_a c_a :: c_c:c_b:c_a -> c_c:c_b:c_a encArg :: cons_c:cons_b:cons_a -> c_c:c_b:c_a cons_c :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a cons_b :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a cons_a :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a encode_c :: cons_c:cons_b:cons_a -> c_c:c_b:c_a encode_b :: cons_c:cons_b:cons_a -> c_c:c_b:c_a encode_a :: cons_c:cons_b:cons_a -> c_c:c_b:c_a Rewrite Strategy: INNERMOST ---------------------------------------- (23) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: encArg_1 encode_c_1 encode_b_1 encode_a_1 a_1 b_1 c_1 Due to the following rules being added: encArg(v0) -> const [0] encode_c(v0) -> const [0] encode_b(v0) -> const [0] encode_a(v0) -> const [0] a(v0) -> const [0] b(v0) -> const [0] c(v0) -> const [0] And the following fresh constants: const, const1 ---------------------------------------- (24) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(c_b(c_c(x1))) -> c(b(a(x1))) [1] c(c_b(c_a(c_a(x1)))) -> a(a(b(c(x1)))) [1] b(c_a(c_a(c_a(x1)))) -> a(a(a(b(x1)))) [1] c(c_c(c_b(c_b(x1)))) -> b(b(c(c(x1)))) [1] encArg(cons_c(x_1)) -> c(encArg(x_1)) [0] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encode_c(x_1) -> c(encArg(x_1)) [0] encode_b(x_1) -> b(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] a(x0) -> c_a(x0) [0] b(x0) -> c_b(x0) [0] c(x0) -> c_c(x0) [0] encArg(v0) -> const [0] encode_c(v0) -> const [0] encode_b(v0) -> const [0] encode_a(v0) -> const [0] a(v0) -> const [0] b(v0) -> const [0] c(v0) -> const [0] The TRS has the following type information: a :: c_c:c_b:c_a:const -> c_c:c_b:c_a:const c_b :: c_c:c_b:c_a:const -> c_c:c_b:c_a:const c_c :: c_c:c_b:c_a:const -> c_c:c_b:c_a:const c :: c_c:c_b:c_a:const -> c_c:c_b:c_a:const b :: c_c:c_b:c_a:const -> c_c:c_b:c_a:const c_a :: c_c:c_b:c_a:const -> c_c:c_b:c_a:const encArg :: cons_c:cons_b:cons_a -> c_c:c_b:c_a:const cons_c :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a cons_b :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a cons_a :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a encode_c :: cons_c:cons_b:cons_a -> c_c:c_b:c_a:const encode_b :: cons_c:cons_b:cons_a -> c_c:c_b:c_a:const encode_a :: cons_c:cons_b:cons_a -> c_c:c_b:c_a:const const :: c_c:c_b:c_a:const const1 :: cons_c:cons_b:cons_a Rewrite Strategy: INNERMOST ---------------------------------------- (25) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (26) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(c_b(c_c(c_b(c_c(x1'))))) -> c(b(c(b(a(x1'))))) [2] a(c_b(c_c(x1))) -> c(b(c_a(x1))) [1] a(c_b(c_c(x1))) -> c(b(const)) [1] c(c_b(c_a(c_a(c_b(c_a(c_a(x1''))))))) -> a(a(b(a(a(b(c(x1''))))))) [2] c(c_b(c_a(c_a(c_c(c_b(c_b(x11))))))) -> a(a(b(b(b(c(c(x11))))))) [2] c(c_b(c_a(c_a(x1)))) -> a(a(b(c_c(x1)))) [1] c(c_b(c_a(c_a(x1)))) -> a(a(b(const))) [1] b(c_a(c_a(c_a(c_a(c_a(c_a(x12))))))) -> a(a(a(a(a(a(b(x12))))))) [2] b(c_a(c_a(c_a(x1)))) -> a(a(a(c_b(x1)))) [1] b(c_a(c_a(c_a(x1)))) -> a(a(a(const))) [1] c(c_c(c_b(c_b(c_b(c_a(c_a(x13))))))) -> b(b(c(a(a(b(c(x13))))))) [2] c(c_c(c_b(c_b(c_c(c_b(c_b(x14))))))) -> b(b(c(b(b(c(c(x14))))))) [2] c(c_c(c_b(c_b(x1)))) -> b(b(c(c_c(x1)))) [1] c(c_c(c_b(c_b(x1)))) -> b(b(c(const))) [1] encArg(cons_c(cons_c(x_1'))) -> c(c(encArg(x_1'))) [0] encArg(cons_c(cons_b(x_1''))) -> c(b(encArg(x_1''))) [0] encArg(cons_c(cons_a(x_11))) -> c(a(encArg(x_11))) [0] encArg(cons_c(x_1)) -> c(const) [0] encArg(cons_b(cons_c(x_12))) -> b(c(encArg(x_12))) [0] encArg(cons_b(cons_b(x_13))) -> b(b(encArg(x_13))) [0] encArg(cons_b(cons_a(x_14))) -> b(a(encArg(x_14))) [0] encArg(cons_b(x_1)) -> b(const) [0] encArg(cons_a(cons_c(x_15))) -> a(c(encArg(x_15))) [0] encArg(cons_a(cons_b(x_16))) -> a(b(encArg(x_16))) [0] encArg(cons_a(cons_a(x_17))) -> a(a(encArg(x_17))) [0] encArg(cons_a(x_1)) -> a(const) [0] encode_c(cons_c(x_18)) -> c(c(encArg(x_18))) [0] encode_c(cons_b(x_19)) -> c(b(encArg(x_19))) [0] encode_c(cons_a(x_110)) -> c(a(encArg(x_110))) [0] encode_c(x_1) -> c(const) [0] encode_b(cons_c(x_111)) -> b(c(encArg(x_111))) [0] encode_b(cons_b(x_112)) -> b(b(encArg(x_112))) [0] encode_b(cons_a(x_113)) -> b(a(encArg(x_113))) [0] encode_b(x_1) -> b(const) [0] encode_a(cons_c(x_114)) -> a(c(encArg(x_114))) [0] encode_a(cons_b(x_115)) -> a(b(encArg(x_115))) [0] encode_a(cons_a(x_116)) -> a(a(encArg(x_116))) [0] encode_a(x_1) -> a(const) [0] a(x0) -> c_a(x0) [0] b(x0) -> c_b(x0) [0] c(x0) -> c_c(x0) [0] encArg(v0) -> const [0] encode_c(v0) -> const [0] encode_b(v0) -> const [0] encode_a(v0) -> const [0] a(v0) -> const [0] b(v0) -> const [0] c(v0) -> const [0] The TRS has the following type information: a :: c_c:c_b:c_a:const -> c_c:c_b:c_a:const c_b :: c_c:c_b:c_a:const -> c_c:c_b:c_a:const c_c :: c_c:c_b:c_a:const -> c_c:c_b:c_a:const c :: c_c:c_b:c_a:const -> c_c:c_b:c_a:const b :: c_c:c_b:c_a:const -> c_c:c_b:c_a:const c_a :: c_c:c_b:c_a:const -> c_c:c_b:c_a:const encArg :: cons_c:cons_b:cons_a -> c_c:c_b:c_a:const cons_c :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a cons_b :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a cons_a :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a encode_c :: cons_c:cons_b:cons_a -> c_c:c_b:c_a:const encode_b :: cons_c:cons_b:cons_a -> c_c:c_b:c_a:const encode_a :: cons_c:cons_b:cons_a -> c_c:c_b:c_a:const const :: c_c:c_b:c_a:const const1 :: cons_c:cons_b:cons_a Rewrite Strategy: INNERMOST ---------------------------------------- (27) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: const => 0 const1 => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 2 }-> c(b(c(b(a(x1'))))) :|: x1' >= 0, z = 1 + (1 + (1 + (1 + x1'))) a(z) -{ 1 }-> c(b(0)) :|: z = 1 + (1 + x1), x1 >= 0 a(z) -{ 1 }-> c(b(1 + x1)) :|: z = 1 + (1 + x1), x1 >= 0 a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 b(z) -{ 2 }-> a(a(a(a(a(a(b(x12))))))) :|: z = 1 + (1 + (1 + (1 + (1 + (1 + x12))))), x12 >= 0 b(z) -{ 1 }-> a(a(a(0))) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) b(z) -{ 1 }-> a(a(a(1 + x1))) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 b(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 c(z) -{ 2 }-> b(b(c(b(b(c(c(x14))))))) :|: z = 1 + (1 + (1 + (1 + (1 + (1 + x14))))), x14 >= 0 c(z) -{ 2 }-> b(b(c(a(a(b(c(x13))))))) :|: x13 >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + x13))))) c(z) -{ 1 }-> b(b(c(0))) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) c(z) -{ 1 }-> b(b(c(1 + x1))) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) c(z) -{ 2 }-> a(a(b(b(b(c(c(x11))))))) :|: x11 >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + x11))))) c(z) -{ 2 }-> a(a(b(a(a(b(c(x1''))))))) :|: x1'' >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + x1''))))) c(z) -{ 1 }-> a(a(b(0))) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) c(z) -{ 1 }-> a(a(b(1 + x1))) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 c(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 encArg(z) -{ 0 }-> c(c(encArg(x_1'))) :|: z = 1 + (1 + x_1'), x_1' >= 0 encArg(z) -{ 0 }-> c(b(encArg(x_1''))) :|: z = 1 + (1 + x_1''), x_1'' >= 0 encArg(z) -{ 0 }-> c(a(encArg(x_11))) :|: x_11 >= 0, z = 1 + (1 + x_11) encArg(z) -{ 0 }-> c(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> b(c(encArg(x_12))) :|: z = 1 + (1 + x_12), x_12 >= 0 encArg(z) -{ 0 }-> b(b(encArg(x_13))) :|: z = 1 + (1 + x_13), x_13 >= 0 encArg(z) -{ 0 }-> b(a(encArg(x_14))) :|: x_14 >= 0, z = 1 + (1 + x_14) encArg(z) -{ 0 }-> b(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> a(c(encArg(x_15))) :|: x_15 >= 0, z = 1 + (1 + x_15) encArg(z) -{ 0 }-> a(b(encArg(x_16))) :|: x_16 >= 0, z = 1 + (1 + x_16) encArg(z) -{ 0 }-> a(a(encArg(x_17))) :|: x_17 >= 0, z = 1 + (1 + x_17) encArg(z) -{ 0 }-> a(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_a(z) -{ 0 }-> a(c(encArg(x_114))) :|: x_114 >= 0, z = 1 + x_114 encode_a(z) -{ 0 }-> a(b(encArg(x_115))) :|: x_115 >= 0, z = 1 + x_115 encode_a(z) -{ 0 }-> a(a(encArg(x_116))) :|: x_116 >= 0, z = 1 + x_116 encode_a(z) -{ 0 }-> a(0) :|: x_1 >= 0, z = x_1 encode_a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_b(z) -{ 0 }-> b(c(encArg(x_111))) :|: z = 1 + x_111, x_111 >= 0 encode_b(z) -{ 0 }-> b(b(encArg(x_112))) :|: z = 1 + x_112, x_112 >= 0 encode_b(z) -{ 0 }-> b(a(encArg(x_113))) :|: z = 1 + x_113, x_113 >= 0 encode_b(z) -{ 0 }-> b(0) :|: x_1 >= 0, z = x_1 encode_b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_c(z) -{ 0 }-> c(c(encArg(x_18))) :|: z = 1 + x_18, x_18 >= 0 encode_c(z) -{ 0 }-> c(b(encArg(x_19))) :|: z = 1 + x_19, x_19 >= 0 encode_c(z) -{ 0 }-> c(a(encArg(x_110))) :|: z = 1 + x_110, x_110 >= 0 encode_c(z) -{ 0 }-> c(0) :|: x_1 >= 0, z = x_1 encode_c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (29) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 2 }-> c(b(c(b(a(z - 4))))) :|: z - 4 >= 0 a(z) -{ 1 }-> c(b(0)) :|: z - 2 >= 0 a(z) -{ 1 }-> c(b(1 + (z - 2))) :|: z - 2 >= 0 a(z) -{ 0 }-> 0 :|: z >= 0 a(z) -{ 0 }-> 1 + z :|: z >= 0 b(z) -{ 2 }-> a(a(a(a(a(a(b(z - 6))))))) :|: z - 6 >= 0 b(z) -{ 1 }-> a(a(a(0))) :|: z - 3 >= 0 b(z) -{ 1 }-> a(a(a(1 + (z - 3)))) :|: z - 3 >= 0 b(z) -{ 0 }-> 0 :|: z >= 0 b(z) -{ 0 }-> 1 + z :|: z >= 0 c(z) -{ 2 }-> b(b(c(b(b(c(c(z - 6))))))) :|: z - 6 >= 0 c(z) -{ 2 }-> b(b(c(a(a(b(c(z - 6))))))) :|: z - 6 >= 0 c(z) -{ 1 }-> b(b(c(0))) :|: z - 3 >= 0 c(z) -{ 1 }-> b(b(c(1 + (z - 3)))) :|: z - 3 >= 0 c(z) -{ 2 }-> a(a(b(b(b(c(c(z - 6))))))) :|: z - 6 >= 0 c(z) -{ 2 }-> a(a(b(a(a(b(c(z - 6))))))) :|: z - 6 >= 0 c(z) -{ 1 }-> a(a(b(0))) :|: z - 3 >= 0 c(z) -{ 1 }-> a(a(b(1 + (z - 3)))) :|: z - 3 >= 0 c(z) -{ 0 }-> 0 :|: z >= 0 c(z) -{ 0 }-> 1 + z :|: z >= 0 encArg(z) -{ 0 }-> c(c(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> c(b(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> c(a(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> c(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> b(c(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> b(b(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> b(a(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> b(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> a(c(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> a(b(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> a(a(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> a(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_a(z) -{ 0 }-> a(c(encArg(z - 1))) :|: z - 1 >= 0 encode_a(z) -{ 0 }-> a(b(encArg(z - 1))) :|: z - 1 >= 0 encode_a(z) -{ 0 }-> a(a(encArg(z - 1))) :|: z - 1 >= 0 encode_a(z) -{ 0 }-> a(0) :|: z >= 0 encode_a(z) -{ 0 }-> 0 :|: z >= 0 encode_b(z) -{ 0 }-> b(c(encArg(z - 1))) :|: z - 1 >= 0 encode_b(z) -{ 0 }-> b(b(encArg(z - 1))) :|: z - 1 >= 0 encode_b(z) -{ 0 }-> b(a(encArg(z - 1))) :|: z - 1 >= 0 encode_b(z) -{ 0 }-> b(0) :|: z >= 0 encode_b(z) -{ 0 }-> 0 :|: z >= 0 encode_c(z) -{ 0 }-> c(c(encArg(z - 1))) :|: z - 1 >= 0 encode_c(z) -{ 0 }-> c(b(encArg(z - 1))) :|: z - 1 >= 0 encode_c(z) -{ 0 }-> c(a(encArg(z - 1))) :|: z - 1 >= 0 encode_c(z) -{ 0 }-> c(0) :|: z >= 0 encode_c(z) -{ 0 }-> 0 :|: z >= 0 ---------------------------------------- (31) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: encArg(v0) -> null_encArg [0] encode_c(v0) -> null_encode_c [0] encode_b(v0) -> null_encode_b [0] encode_a(v0) -> null_encode_a [0] a(v0) -> null_a [0] b(v0) -> null_b [0] c(v0) -> null_c [0] And the following fresh constants: null_encArg, null_encode_c, null_encode_b, null_encode_a, null_a, null_b, null_c, const ---------------------------------------- (32) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(c_b(c_c(x1))) -> c(b(a(x1))) [1] c(c_b(c_a(c_a(x1)))) -> a(a(b(c(x1)))) [1] b(c_a(c_a(c_a(x1)))) -> a(a(a(b(x1)))) [1] c(c_c(c_b(c_b(x1)))) -> b(b(c(c(x1)))) [1] encArg(cons_c(x_1)) -> c(encArg(x_1)) [0] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encode_c(x_1) -> c(encArg(x_1)) [0] encode_b(x_1) -> b(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] a(x0) -> c_a(x0) [0] b(x0) -> c_b(x0) [0] c(x0) -> c_c(x0) [0] encArg(v0) -> null_encArg [0] encode_c(v0) -> null_encode_c [0] encode_b(v0) -> null_encode_b [0] encode_a(v0) -> null_encode_a [0] a(v0) -> null_a [0] b(v0) -> null_b [0] c(v0) -> null_c [0] The TRS has the following type information: a :: c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c -> c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c c_b :: c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c -> c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c c_c :: c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c -> c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c c :: c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c -> c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c b :: c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c -> c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c c_a :: c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c -> c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c encArg :: cons_c:cons_b:cons_a -> c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c cons_c :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a cons_b :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a cons_a :: cons_c:cons_b:cons_a -> cons_c:cons_b:cons_a encode_c :: cons_c:cons_b:cons_a -> c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c encode_b :: cons_c:cons_b:cons_a -> c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c encode_a :: cons_c:cons_b:cons_a -> c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c null_encArg :: c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c null_encode_c :: c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c null_encode_b :: c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c null_encode_a :: c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c null_a :: c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c null_b :: c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c null_c :: c_c:c_b:c_a:null_encArg:null_encode_c:null_encode_b:null_encode_a:null_a:null_b:null_c const :: cons_c:cons_b:cons_a Rewrite Strategy: INNERMOST ---------------------------------------- (33) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: null_encArg => 0 null_encode_c => 0 null_encode_b => 0 null_encode_a => 0 null_a => 0 null_b => 0 null_c => 0 const => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 1 }-> c(b(a(x1))) :|: z = 1 + (1 + x1), x1 >= 0 a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 b(z) -{ 1 }-> a(a(a(b(x1)))) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 b(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 c(z) -{ 1 }-> b(b(c(c(x1)))) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) c(z) -{ 1 }-> a(a(b(c(x1)))) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 c(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 encArg(z) -{ 0 }-> c(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> b(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> a(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_a(z) -{ 0 }-> a(encArg(x_1)) :|: x_1 >= 0, z = x_1 encode_a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_b(z) -{ 0 }-> b(encArg(x_1)) :|: x_1 >= 0, z = x_1 encode_b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_c(z) -{ 0 }-> c(encArg(x_1)) :|: x_1 >= 0, z = x_1 encode_c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encode_c(z0) -> c(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: ENCARG(cons_c(z0)) -> c1(C(encArg(z0)), ENCARG(z0)) ENCARG(cons_b(z0)) -> c2(B(encArg(z0)), ENCARG(z0)) ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0)) ENCODE_C(z0) -> c4(C(encArg(z0)), ENCARG(z0)) ENCODE_B(z0) -> c5(B(encArg(z0)), ENCARG(z0)) ENCODE_A(z0) -> c6(A(encArg(z0)), ENCARG(z0)) C(b(a(a(z0)))) -> c7(A(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) S tuples: C(b(a(a(z0)))) -> c7(A(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: c_1, b_1, a_1, encArg_1, encode_c_1, encode_b_1, encode_a_1 Defined Pair Symbols: ENCARG_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, C_1, B_1, A_1 Compound Symbols: c1_2, c2_2, c3_2, c4_2, c5_2, c6_2, c7_4, c8_4, c9_4, c10_3 ---------------------------------------- (37) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encode_c(z0) -> c(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: ENCARG(cons_c(z0)) -> c1(C(encArg(z0)), ENCARG(z0)) ENCARG(cons_b(z0)) -> c2(B(encArg(z0)), ENCARG(z0)) ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0)) C(b(a(a(z0)))) -> c7(A(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_C(z0) -> c11(C(encArg(z0))) ENCODE_C(z0) -> c11(ENCARG(z0)) ENCODE_B(z0) -> c11(B(encArg(z0))) ENCODE_B(z0) -> c11(ENCARG(z0)) ENCODE_A(z0) -> c11(A(encArg(z0))) ENCODE_A(z0) -> c11(ENCARG(z0)) S tuples: C(b(a(a(z0)))) -> c7(A(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: c_1, b_1, a_1, encArg_1, encode_c_1, encode_b_1, encode_a_1 Defined Pair Symbols: ENCARG_1, C_1, B_1, A_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c1_2, c2_2, c3_2, c7_4, c8_4, c9_4, c10_3, c11_1 ---------------------------------------- (39) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 3 leading nodes: ENCODE_C(z0) -> c11(ENCARG(z0)) ENCODE_B(z0) -> c11(ENCARG(z0)) ENCODE_A(z0) -> c11(ENCARG(z0)) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encode_c(z0) -> c(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: ENCARG(cons_c(z0)) -> c1(C(encArg(z0)), ENCARG(z0)) ENCARG(cons_b(z0)) -> c2(B(encArg(z0)), ENCARG(z0)) ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0)) C(b(a(a(z0)))) -> c7(A(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_C(z0) -> c11(C(encArg(z0))) ENCODE_B(z0) -> c11(B(encArg(z0))) ENCODE_A(z0) -> c11(A(encArg(z0))) S tuples: C(b(a(a(z0)))) -> c7(A(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: c_1, b_1, a_1, encArg_1, encode_c_1, encode_b_1, encode_a_1 Defined Pair Symbols: ENCARG_1, C_1, B_1, A_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c1_2, c2_2, c3_2, c7_4, c8_4, c9_4, c10_3, c11_1 ---------------------------------------- (41) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_c(z0) -> c(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: ENCARG(cons_c(z0)) -> c1(C(encArg(z0)), ENCARG(z0)) ENCARG(cons_b(z0)) -> c2(B(encArg(z0)), ENCARG(z0)) ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0)) C(b(a(a(z0)))) -> c7(A(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_C(z0) -> c11(C(encArg(z0))) ENCODE_B(z0) -> c11(B(encArg(z0))) ENCODE_A(z0) -> c11(A(encArg(z0))) S tuples: C(b(a(a(z0)))) -> c7(A(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: ENCARG_1, C_1, B_1, A_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c1_2, c2_2, c3_2, c7_4, c8_4, c9_4, c10_3, c11_1 ---------------------------------------- (43) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_c(z0)) -> c1(C(encArg(z0)), ENCARG(z0)) by ENCARG(cons_c(cons_c(z0))) -> c1(C(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_c(cons_b(z0))) -> c1(C(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: ENCARG(cons_b(z0)) -> c2(B(encArg(z0)), ENCARG(z0)) ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0)) C(b(a(a(z0)))) -> c7(A(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_C(z0) -> c11(C(encArg(z0))) ENCODE_B(z0) -> c11(B(encArg(z0))) ENCODE_A(z0) -> c11(A(encArg(z0))) ENCARG(cons_c(cons_c(z0))) -> c1(C(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_c(cons_b(z0))) -> c1(C(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) S tuples: C(b(a(a(z0)))) -> c7(A(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: ENCARG_1, C_1, B_1, A_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c2_2, c3_2, c7_4, c8_4, c9_4, c10_3, c11_1, c1_2 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(z0)) -> c2(B(encArg(z0)), ENCARG(z0)) by ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0)) C(b(a(a(z0)))) -> c7(A(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_C(z0) -> c11(C(encArg(z0))) ENCODE_B(z0) -> c11(B(encArg(z0))) ENCODE_A(z0) -> c11(A(encArg(z0))) ENCARG(cons_c(cons_c(z0))) -> c1(C(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_c(cons_b(z0))) -> c1(C(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) S tuples: C(b(a(a(z0)))) -> c7(A(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: ENCARG_1, C_1, B_1, A_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c3_2, c7_4, c8_4, c9_4, c10_3, c11_1, c1_2, c2_2 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_a(z0)) -> c3(A(encArg(z0)), ENCARG(z0)) by ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: C(b(a(a(z0)))) -> c7(A(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_C(z0) -> c11(C(encArg(z0))) ENCODE_B(z0) -> c11(B(encArg(z0))) ENCODE_A(z0) -> c11(A(encArg(z0))) ENCARG(cons_c(cons_c(z0))) -> c1(C(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_c(cons_b(z0))) -> c1(C(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) S tuples: C(b(a(a(z0)))) -> c7(A(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: C_1, B_1, A_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1 Compound Symbols: c7_4, c8_4, c9_4, c10_3, c11_1, c1_2, c2_2, c3_2 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(b(a(a(z0)))) -> c7(A(a(b(c(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) by C(b(a(a(z0)))) -> c7(A(c(b(a(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_C(z0) -> c11(C(encArg(z0))) ENCODE_B(z0) -> c11(B(encArg(z0))) ENCODE_A(z0) -> c11(A(encArg(z0))) ENCARG(cons_c(cons_c(z0))) -> c1(C(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_c(cons_b(z0))) -> c1(C(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(z0)))) -> c7(A(c(b(a(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) S tuples: C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(z0)))) -> c7(A(c(b(a(z0)))), A(b(c(z0))), B(c(z0)), C(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: C_1, B_1, A_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1 Compound Symbols: c8_4, c9_4, c10_3, c11_1, c1_2, c2_2, c3_2, c7_4 ---------------------------------------- (51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_C(z0) -> c11(C(encArg(z0))) ENCODE_B(z0) -> c11(B(encArg(z0))) ENCODE_A(z0) -> c11(A(encArg(z0))) ENCARG(cons_c(cons_c(z0))) -> c1(C(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_c(cons_b(z0))) -> c1(C(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) S tuples: C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: C_1, B_1, A_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1 Compound Symbols: c8_4, c9_4, c10_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_3 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(c(b(b(z0)))) -> c8(B(b(c(c(z0)))), B(c(c(z0))), C(c(z0)), C(z0)) by C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0)))), C(b(b(z0)))) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_C(z0) -> c11(C(encArg(z0))) ENCODE_B(z0) -> c11(B(encArg(z0))) ENCODE_A(z0) -> c11(A(encArg(z0))) ENCARG(cons_c(cons_c(z0))) -> c1(C(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_c(cons_b(z0))) -> c1(C(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0)))), C(b(b(z0)))) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) S tuples: B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0)))), C(b(b(z0)))) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: B_1, A_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, C_1 Compound Symbols: c9_4, c10_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_3, c8_4 ---------------------------------------- (55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_C(z0) -> c11(C(encArg(z0))) ENCODE_B(z0) -> c11(B(encArg(z0))) ENCODE_A(z0) -> c11(A(encArg(z0))) ENCARG(cons_c(cons_c(z0))) -> c1(C(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_c(cons_b(z0))) -> c1(C(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) S tuples: B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: B_1, A_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, C_1 Compound Symbols: c9_4, c10_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_3, c8_4, c8_3 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(a(a(a(z0)))) -> c9(A(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0)) by B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0))), B(c(z0))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_C(z0) -> c11(C(encArg(z0))) ENCODE_B(z0) -> c11(B(encArg(z0))) ENCODE_A(z0) -> c11(A(encArg(z0))) ENCARG(cons_c(cons_c(z0))) -> c1(C(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_c(cons_b(z0))) -> c1(C(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0))), B(c(z0))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0))), B(c(z0))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, C_1, B_1 Compound Symbols: c10_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_3, c8_4, c8_3, c9_4 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_C(z0) -> c11(C(encArg(z0))) ENCODE_B(z0) -> c11(B(encArg(z0))) ENCODE_A(z0) -> c11(A(encArg(z0))) ENCARG(cons_c(cons_c(z0))) -> c1(C(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_c(cons_b(z0))) -> c1(C(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, C_1, B_1 Compound Symbols: c10_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_3, c8_4, c8_3, c9_4, c9_3 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_C(z0) -> c11(C(encArg(z0))) by ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_B(z0) -> c11(B(encArg(z0))) ENCODE_A(z0) -> c11(A(encArg(z0))) ENCARG(cons_c(cons_c(z0))) -> c1(C(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_c(cons_b(z0))) -> c1(C(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, C_1, B_1, ENCODE_C_1 Compound Symbols: c10_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_3, c8_4, c8_3, c9_4, c9_3 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_B(z0) -> c11(B(encArg(z0))) by ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_A(z0) -> c11(A(encArg(z0))) ENCARG(cons_c(cons_c(z0))) -> c1(C(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_c(cons_b(z0))) -> c1(C(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCODE_A_1, ENCARG_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1 Compound Symbols: c10_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_3, c8_4, c8_3, c9_4, c9_3 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_A(z0) -> c11(A(encArg(z0))) by ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCARG(cons_c(cons_c(z0))) -> c1(C(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_c(cons_b(z0))) -> c1(C(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCARG_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c10_3, c1_2, c2_2, c3_2, c7_4, c7_3, c8_4, c8_3, c9_4, c9_3, c11_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_c(cons_c(z0))) -> c1(C(c(encArg(z0))), ENCARG(cons_c(z0))) by ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCARG(cons_c(cons_b(z0))) -> c1(C(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCARG_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c10_3, c1_2, c2_2, c3_2, c7_4, c7_3, c8_4, c8_3, c9_4, c9_3, c11_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_c(cons_b(z0))) -> c1(C(b(encArg(z0))), ENCARG(cons_b(z0))) by ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCARG_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c10_3, c1_2, c2_2, c3_2, c7_4, c7_3, c8_4, c8_3, c9_4, c9_3, c11_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_c(cons_a(z0))) -> c1(C(a(encArg(z0))), ENCARG(cons_a(z0))) by ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCARG_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c10_3, c2_2, c3_2, c7_4, c7_3, c8_4, c8_3, c9_4, c9_3, c11_1, c1_2 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(cons_c(z0))) -> c2(B(c(encArg(z0))), ENCARG(cons_c(z0))) by ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCARG_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c10_3, c2_2, c3_2, c7_4, c7_3, c8_4, c8_3, c9_4, c9_3, c11_1, c1_2 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(cons_b(z0))) -> c2(B(b(encArg(z0))), ENCARG(cons_b(z0))) by ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCARG_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c10_3, c2_2, c3_2, c7_4, c7_3, c8_4, c8_3, c9_4, c9_3, c11_1, c1_2 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(cons_a(z0))) -> c2(B(a(encArg(z0))), ENCARG(cons_a(z0))) by ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCARG_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c10_3, c3_2, c7_4, c7_3, c8_4, c8_3, c9_4, c9_3, c11_1, c1_2, c2_2 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_a(cons_c(z0))) -> c3(A(c(encArg(z0))), ENCARG(cons_c(z0))) by ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCARG_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c10_3, c3_2, c7_4, c7_3, c8_4, c8_3, c9_4, c9_3, c11_1, c1_2, c2_2 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_a(cons_b(z0))) -> c3(A(b(encArg(z0))), ENCARG(cons_b(z0))) by ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCARG_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c10_3, c3_2, c7_4, c7_3, c8_4, c8_3, c9_4, c9_3, c11_1, c1_2, c2_2 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_a(cons_a(z0))) -> c3(A(a(encArg(z0))), ENCARG(cons_a(z0))) by ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1 Compound Symbols: c10_3, c7_4, c7_3, c8_4, c8_3, c9_4, c9_3, c11_1, c1_2, c2_2, c3_2 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(a(b(c(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) by C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(c(b(a(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(c(b(a(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) C(b(a(a(b(a(a(z0))))))) -> c7(A(a(b(a(c(b(a(z0))))))), A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1 Compound Symbols: c10_3, c7_4, c7_3, c8_4, c8_3, c9_4, c9_3, c11_1, c1_2, c2_2, c3_2, c7_1 ---------------------------------------- (87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1 Compound Symbols: c10_3, c7_4, c7_3, c8_4, c8_3, c9_4, c9_3, c11_1, c1_2, c2_2, c3_2, c7_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(b(a(a(c(b(b(z0))))))) -> c7(A(a(b(b(b(c(c(z0))))))), A(b(c(c(b(b(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) by C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1 Compound Symbols: c10_3, c7_3, c8_4, c8_3, c9_4, c9_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_1, c7_2 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(b(a(a(z0)))) -> c7(A(b(c(z0))), B(c(z0)), C(z0)) by C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1 Compound Symbols: c10_3, c8_4, c8_3, c9_4, c9_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_1, c7_3, c7_2 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(a(b(c(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) by C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(c(b(a(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(c(b(a(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(z0))))))) -> c8(B(b(c(a(c(b(a(z0))))))), B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1 Compound Symbols: c10_3, c8_4, c8_3, c9_4, c9_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_1, c7_3, c7_2 ---------------------------------------- (95) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1 Compound Symbols: c10_3, c8_4, c8_3, c9_4, c9_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_1, c7_3, c7_2 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(c(b(b(c(b(b(z0))))))) -> c8(B(b(c(b(b(c(c(z0))))))), B(c(c(c(b(b(z0)))))), C(c(c(b(b(z0))))), C(c(b(b(z0))))) by C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, C_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1 Compound Symbols: c10_3, c8_3, c9_4, c9_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_1, c7_3, c7_2, c8_4, c8_2 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(c(b(b(b(b(z0)))))) -> c8(B(b(b(b(c(c(z0)))))), B(c(c(b(b(z0))))), C(c(b(b(z0))))) by C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, C_1 Compound Symbols: c10_3, c9_4, c9_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_1, c7_3, c7_2, c8_4, c8_3, c8_2, c8_1 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(a(a(a(a(a(a(z0))))))) -> c9(A(a(a(a(a(a(b(z0))))))), A(a(b(a(a(a(z0)))))), A(b(a(a(a(z0))))), B(a(a(a(z0))))) by B(a(a(a(a(a(a(c(z0)))))))) -> c9(A(a(a(a(a(c(b(a(z0)))))))), A(a(b(a(a(a(c(z0))))))), A(b(a(a(a(c(z0)))))), B(a(a(a(c(z0)))))) B(a(a(a(a(a(a(a(a(a(z0)))))))))) -> c9(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(a(a(z0))))))))), A(b(a(a(a(a(a(a(z0)))))))), B(a(a(a(a(a(a(z0)))))))) B(a(a(a(a(a(a(x0))))))) -> c9(A(a(b(a(a(a(x0)))))), A(b(a(a(a(x0)))))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) B(a(a(a(a(a(a(c(z0)))))))) -> c9(A(a(a(a(a(c(b(a(z0)))))))), A(a(b(a(a(a(c(z0))))))), A(b(a(a(a(c(z0)))))), B(a(a(a(c(z0)))))) B(a(a(a(a(a(a(a(a(a(z0)))))))))) -> c9(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(a(a(z0))))))))), A(b(a(a(a(a(a(a(z0)))))))), B(a(a(a(a(a(a(z0)))))))) B(a(a(a(a(a(a(x0))))))) -> c9(A(a(b(a(a(a(x0)))))), A(b(a(a(a(x0)))))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) B(a(a(a(a(a(a(c(z0)))))))) -> c9(A(a(a(a(a(c(b(a(z0)))))))), A(a(b(a(a(a(c(z0))))))), A(b(a(a(a(c(z0)))))), B(a(a(a(c(z0)))))) B(a(a(a(a(a(a(a(a(a(z0)))))))))) -> c9(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(a(a(z0))))))))), A(b(a(a(a(a(a(a(z0)))))))), B(a(a(a(a(a(a(z0)))))))) B(a(a(a(a(a(a(x0))))))) -> c9(A(a(b(a(a(a(x0)))))), A(b(a(a(a(x0)))))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, B_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, C_1 Compound Symbols: c10_3, c9_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_1, c7_3, c7_2, c8_4, c8_3, c8_2, c8_1, c9_4, c9_2 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(a(a(a(c(z0))))) -> c9(A(a(c(b(a(z0))))), A(a(b(c(z0)))), A(b(c(z0)))) by B(a(a(a(c(a(z0)))))) -> c9(A(a(a(a(b(c(z0)))))), A(a(b(c(a(z0))))), A(b(c(a(z0))))) B(a(a(a(c(a(a(z0))))))) -> c9(A(a(c(a(a(a(b(z0))))))), A(a(b(c(a(a(z0)))))), A(b(c(a(a(z0)))))) B(a(a(a(c(b(c(z0))))))) -> c9(A(a(c(b(c(b(a(z0))))))), A(a(b(c(b(c(z0)))))), A(b(c(b(c(z0)))))) B(a(a(a(c(x0))))) -> c9(A(a(b(c(x0)))), A(b(c(x0)))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) B(a(a(a(a(a(a(c(z0)))))))) -> c9(A(a(a(a(a(c(b(a(z0)))))))), A(a(b(a(a(a(c(z0))))))), A(b(a(a(a(c(z0)))))), B(a(a(a(c(z0)))))) B(a(a(a(a(a(a(a(a(a(z0)))))))))) -> c9(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(a(a(z0))))))))), A(b(a(a(a(a(a(a(z0)))))))), B(a(a(a(a(a(a(z0)))))))) B(a(a(a(a(a(a(x0))))))) -> c9(A(a(b(a(a(a(x0)))))), A(b(a(a(a(x0)))))) B(a(a(a(c(a(z0)))))) -> c9(A(a(a(a(b(c(z0)))))), A(a(b(c(a(z0))))), A(b(c(a(z0))))) B(a(a(a(c(a(a(z0))))))) -> c9(A(a(c(a(a(a(b(z0))))))), A(a(b(c(a(a(z0)))))), A(b(c(a(a(z0)))))) B(a(a(a(c(b(c(z0))))))) -> c9(A(a(c(b(c(b(a(z0))))))), A(a(b(c(b(c(z0)))))), A(b(c(b(c(z0)))))) B(a(a(a(c(x0))))) -> c9(A(a(b(c(x0)))), A(b(c(x0)))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) B(a(a(a(a(a(a(c(z0)))))))) -> c9(A(a(a(a(a(c(b(a(z0)))))))), A(a(b(a(a(a(c(z0))))))), A(b(a(a(a(c(z0)))))), B(a(a(a(c(z0)))))) B(a(a(a(a(a(a(a(a(a(z0)))))))))) -> c9(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(a(a(z0))))))))), A(b(a(a(a(a(a(a(z0)))))))), B(a(a(a(a(a(a(z0)))))))) B(a(a(a(a(a(a(x0))))))) -> c9(A(a(b(a(a(a(x0)))))), A(b(a(a(a(x0)))))) B(a(a(a(c(a(z0)))))) -> c9(A(a(a(a(b(c(z0)))))), A(a(b(c(a(z0))))), A(b(c(a(z0))))) B(a(a(a(c(a(a(z0))))))) -> c9(A(a(c(a(a(a(b(z0))))))), A(a(b(c(a(a(z0)))))), A(b(c(a(a(z0)))))) B(a(a(a(c(b(c(z0))))))) -> c9(A(a(c(b(c(b(a(z0))))))), A(a(b(c(b(c(z0)))))), A(b(c(b(c(z0)))))) B(a(a(a(c(x0))))) -> c9(A(a(b(c(x0)))), A(b(c(x0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, C_1, B_1 Compound Symbols: c10_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_1, c7_3, c7_2, c8_4, c8_3, c8_2, c8_1, c9_4, c9_2, c9_3 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_C(cons_c(z0)) -> c11(C(c(encArg(z0)))) by ENCODE_C(cons_c(cons_c(z0))) -> c11(C(c(c(encArg(z0))))) ENCODE_C(cons_c(cons_b(z0))) -> c11(C(c(b(encArg(z0))))) ENCODE_C(cons_c(cons_a(z0))) -> c11(C(c(a(encArg(z0))))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) B(a(a(a(a(a(a(c(z0)))))))) -> c9(A(a(a(a(a(c(b(a(z0)))))))), A(a(b(a(a(a(c(z0))))))), A(b(a(a(a(c(z0)))))), B(a(a(a(c(z0)))))) B(a(a(a(a(a(a(a(a(a(z0)))))))))) -> c9(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(a(a(z0))))))))), A(b(a(a(a(a(a(a(z0)))))))), B(a(a(a(a(a(a(z0)))))))) B(a(a(a(a(a(a(x0))))))) -> c9(A(a(b(a(a(a(x0)))))), A(b(a(a(a(x0)))))) B(a(a(a(c(a(z0)))))) -> c9(A(a(a(a(b(c(z0)))))), A(a(b(c(a(z0))))), A(b(c(a(z0))))) B(a(a(a(c(a(a(z0))))))) -> c9(A(a(c(a(a(a(b(z0))))))), A(a(b(c(a(a(z0)))))), A(b(c(a(a(z0)))))) B(a(a(a(c(b(c(z0))))))) -> c9(A(a(c(b(c(b(a(z0))))))), A(a(b(c(b(c(z0)))))), A(b(c(b(c(z0)))))) B(a(a(a(c(x0))))) -> c9(A(a(b(c(x0)))), A(b(c(x0)))) ENCODE_C(cons_c(cons_c(z0))) -> c11(C(c(c(encArg(z0))))) ENCODE_C(cons_c(cons_b(z0))) -> c11(C(c(b(encArg(z0))))) ENCODE_C(cons_c(cons_a(z0))) -> c11(C(c(a(encArg(z0))))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) B(a(a(a(a(a(a(c(z0)))))))) -> c9(A(a(a(a(a(c(b(a(z0)))))))), A(a(b(a(a(a(c(z0))))))), A(b(a(a(a(c(z0)))))), B(a(a(a(c(z0)))))) B(a(a(a(a(a(a(a(a(a(z0)))))))))) -> c9(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(a(a(z0))))))))), A(b(a(a(a(a(a(a(z0)))))))), B(a(a(a(a(a(a(z0)))))))) B(a(a(a(a(a(a(x0))))))) -> c9(A(a(b(a(a(a(x0)))))), A(b(a(a(a(x0)))))) B(a(a(a(c(a(z0)))))) -> c9(A(a(a(a(b(c(z0)))))), A(a(b(c(a(z0))))), A(b(c(a(z0))))) B(a(a(a(c(a(a(z0))))))) -> c9(A(a(c(a(a(a(b(z0))))))), A(a(b(c(a(a(z0)))))), A(b(c(a(a(z0)))))) B(a(a(a(c(b(c(z0))))))) -> c9(A(a(c(b(c(b(a(z0))))))), A(a(b(c(b(c(z0)))))), A(b(c(b(c(z0)))))) B(a(a(a(c(x0))))) -> c9(A(a(b(c(x0)))), A(b(c(x0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, C_1, B_1 Compound Symbols: c10_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_1, c7_3, c7_2, c8_4, c8_3, c8_2, c8_1, c9_4, c9_2, c9_3 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_C(cons_b(z0)) -> c11(C(b(encArg(z0)))) by ENCODE_C(cons_b(cons_c(z0))) -> c11(C(b(c(encArg(z0))))) ENCODE_C(cons_b(cons_b(z0))) -> c11(C(b(b(encArg(z0))))) ENCODE_C(cons_b(cons_a(z0))) -> c11(C(b(a(encArg(z0))))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) B(a(a(a(a(a(a(c(z0)))))))) -> c9(A(a(a(a(a(c(b(a(z0)))))))), A(a(b(a(a(a(c(z0))))))), A(b(a(a(a(c(z0)))))), B(a(a(a(c(z0)))))) B(a(a(a(a(a(a(a(a(a(z0)))))))))) -> c9(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(a(a(z0))))))))), A(b(a(a(a(a(a(a(z0)))))))), B(a(a(a(a(a(a(z0)))))))) B(a(a(a(a(a(a(x0))))))) -> c9(A(a(b(a(a(a(x0)))))), A(b(a(a(a(x0)))))) B(a(a(a(c(a(z0)))))) -> c9(A(a(a(a(b(c(z0)))))), A(a(b(c(a(z0))))), A(b(c(a(z0))))) B(a(a(a(c(a(a(z0))))))) -> c9(A(a(c(a(a(a(b(z0))))))), A(a(b(c(a(a(z0)))))), A(b(c(a(a(z0)))))) B(a(a(a(c(b(c(z0))))))) -> c9(A(a(c(b(c(b(a(z0))))))), A(a(b(c(b(c(z0)))))), A(b(c(b(c(z0)))))) B(a(a(a(c(x0))))) -> c9(A(a(b(c(x0)))), A(b(c(x0)))) ENCODE_C(cons_c(cons_c(z0))) -> c11(C(c(c(encArg(z0))))) ENCODE_C(cons_c(cons_b(z0))) -> c11(C(c(b(encArg(z0))))) ENCODE_C(cons_c(cons_a(z0))) -> c11(C(c(a(encArg(z0))))) ENCODE_C(cons_b(cons_c(z0))) -> c11(C(b(c(encArg(z0))))) ENCODE_C(cons_b(cons_b(z0))) -> c11(C(b(b(encArg(z0))))) ENCODE_C(cons_b(cons_a(z0))) -> c11(C(b(a(encArg(z0))))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) B(a(a(a(a(a(a(c(z0)))))))) -> c9(A(a(a(a(a(c(b(a(z0)))))))), A(a(b(a(a(a(c(z0))))))), A(b(a(a(a(c(z0)))))), B(a(a(a(c(z0)))))) B(a(a(a(a(a(a(a(a(a(z0)))))))))) -> c9(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(a(a(z0))))))))), A(b(a(a(a(a(a(a(z0)))))))), B(a(a(a(a(a(a(z0)))))))) B(a(a(a(a(a(a(x0))))))) -> c9(A(a(b(a(a(a(x0)))))), A(b(a(a(a(x0)))))) B(a(a(a(c(a(z0)))))) -> c9(A(a(a(a(b(c(z0)))))), A(a(b(c(a(z0))))), A(b(c(a(z0))))) B(a(a(a(c(a(a(z0))))))) -> c9(A(a(c(a(a(a(b(z0))))))), A(a(b(c(a(a(z0)))))), A(b(c(a(a(z0)))))) B(a(a(a(c(b(c(z0))))))) -> c9(A(a(c(b(c(b(a(z0))))))), A(a(b(c(b(c(z0)))))), A(b(c(b(c(z0)))))) B(a(a(a(c(x0))))) -> c9(A(a(b(c(x0)))), A(b(c(x0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCODE_C_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, C_1, B_1 Compound Symbols: c10_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_1, c7_3, c7_2, c8_4, c8_3, c8_2, c8_1, c9_4, c9_2, c9_3 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_C(cons_a(z0)) -> c11(C(a(encArg(z0)))) by ENCODE_C(cons_a(cons_c(z0))) -> c11(C(a(c(encArg(z0))))) ENCODE_C(cons_a(cons_b(z0))) -> c11(C(a(b(encArg(z0))))) ENCODE_C(cons_a(cons_a(z0))) -> c11(C(a(a(encArg(z0))))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) B(a(a(a(a(a(a(c(z0)))))))) -> c9(A(a(a(a(a(c(b(a(z0)))))))), A(a(b(a(a(a(c(z0))))))), A(b(a(a(a(c(z0)))))), B(a(a(a(c(z0)))))) B(a(a(a(a(a(a(a(a(a(z0)))))))))) -> c9(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(a(a(z0))))))))), A(b(a(a(a(a(a(a(z0)))))))), B(a(a(a(a(a(a(z0)))))))) B(a(a(a(a(a(a(x0))))))) -> c9(A(a(b(a(a(a(x0)))))), A(b(a(a(a(x0)))))) B(a(a(a(c(a(z0)))))) -> c9(A(a(a(a(b(c(z0)))))), A(a(b(c(a(z0))))), A(b(c(a(z0))))) B(a(a(a(c(a(a(z0))))))) -> c9(A(a(c(a(a(a(b(z0))))))), A(a(b(c(a(a(z0)))))), A(b(c(a(a(z0)))))) B(a(a(a(c(b(c(z0))))))) -> c9(A(a(c(b(c(b(a(z0))))))), A(a(b(c(b(c(z0)))))), A(b(c(b(c(z0)))))) B(a(a(a(c(x0))))) -> c9(A(a(b(c(x0)))), A(b(c(x0)))) ENCODE_C(cons_c(cons_c(z0))) -> c11(C(c(c(encArg(z0))))) ENCODE_C(cons_c(cons_b(z0))) -> c11(C(c(b(encArg(z0))))) ENCODE_C(cons_c(cons_a(z0))) -> c11(C(c(a(encArg(z0))))) ENCODE_C(cons_b(cons_c(z0))) -> c11(C(b(c(encArg(z0))))) ENCODE_C(cons_b(cons_b(z0))) -> c11(C(b(b(encArg(z0))))) ENCODE_C(cons_b(cons_a(z0))) -> c11(C(b(a(encArg(z0))))) ENCODE_C(cons_a(cons_c(z0))) -> c11(C(a(c(encArg(z0))))) ENCODE_C(cons_a(cons_b(z0))) -> c11(C(a(b(encArg(z0))))) ENCODE_C(cons_a(cons_a(z0))) -> c11(C(a(a(encArg(z0))))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) B(a(a(a(a(a(a(c(z0)))))))) -> c9(A(a(a(a(a(c(b(a(z0)))))))), A(a(b(a(a(a(c(z0))))))), A(b(a(a(a(c(z0)))))), B(a(a(a(c(z0)))))) B(a(a(a(a(a(a(a(a(a(z0)))))))))) -> c9(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(a(a(z0))))))))), A(b(a(a(a(a(a(a(z0)))))))), B(a(a(a(a(a(a(z0)))))))) B(a(a(a(a(a(a(x0))))))) -> c9(A(a(b(a(a(a(x0)))))), A(b(a(a(a(x0)))))) B(a(a(a(c(a(z0)))))) -> c9(A(a(a(a(b(c(z0)))))), A(a(b(c(a(z0))))), A(b(c(a(z0))))) B(a(a(a(c(a(a(z0))))))) -> c9(A(a(c(a(a(a(b(z0))))))), A(a(b(c(a(a(z0)))))), A(b(c(a(a(z0)))))) B(a(a(a(c(b(c(z0))))))) -> c9(A(a(c(b(c(b(a(z0))))))), A(a(b(c(b(c(z0)))))), A(b(c(b(c(z0)))))) B(a(a(a(c(x0))))) -> c9(A(a(b(c(x0)))), A(b(c(x0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, C_1, B_1, ENCODE_C_1 Compound Symbols: c10_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_1, c7_3, c7_2, c8_4, c8_3, c8_2, c8_1, c9_4, c9_2, c9_3 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_B(cons_c(z0)) -> c11(B(c(encArg(z0)))) by ENCODE_B(cons_c(cons_c(z0))) -> c11(B(c(c(encArg(z0))))) ENCODE_B(cons_c(cons_b(z0))) -> c11(B(c(b(encArg(z0))))) ENCODE_B(cons_c(cons_a(z0))) -> c11(B(c(a(encArg(z0))))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) B(a(a(a(a(a(a(c(z0)))))))) -> c9(A(a(a(a(a(c(b(a(z0)))))))), A(a(b(a(a(a(c(z0))))))), A(b(a(a(a(c(z0)))))), B(a(a(a(c(z0)))))) B(a(a(a(a(a(a(a(a(a(z0)))))))))) -> c9(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(a(a(z0))))))))), A(b(a(a(a(a(a(a(z0)))))))), B(a(a(a(a(a(a(z0)))))))) B(a(a(a(a(a(a(x0))))))) -> c9(A(a(b(a(a(a(x0)))))), A(b(a(a(a(x0)))))) B(a(a(a(c(a(z0)))))) -> c9(A(a(a(a(b(c(z0)))))), A(a(b(c(a(z0))))), A(b(c(a(z0))))) B(a(a(a(c(a(a(z0))))))) -> c9(A(a(c(a(a(a(b(z0))))))), A(a(b(c(a(a(z0)))))), A(b(c(a(a(z0)))))) B(a(a(a(c(b(c(z0))))))) -> c9(A(a(c(b(c(b(a(z0))))))), A(a(b(c(b(c(z0)))))), A(b(c(b(c(z0)))))) B(a(a(a(c(x0))))) -> c9(A(a(b(c(x0)))), A(b(c(x0)))) ENCODE_C(cons_c(cons_c(z0))) -> c11(C(c(c(encArg(z0))))) ENCODE_C(cons_c(cons_b(z0))) -> c11(C(c(b(encArg(z0))))) ENCODE_C(cons_c(cons_a(z0))) -> c11(C(c(a(encArg(z0))))) ENCODE_C(cons_b(cons_c(z0))) -> c11(C(b(c(encArg(z0))))) ENCODE_C(cons_b(cons_b(z0))) -> c11(C(b(b(encArg(z0))))) ENCODE_C(cons_b(cons_a(z0))) -> c11(C(b(a(encArg(z0))))) ENCODE_C(cons_a(cons_c(z0))) -> c11(C(a(c(encArg(z0))))) ENCODE_C(cons_a(cons_b(z0))) -> c11(C(a(b(encArg(z0))))) ENCODE_C(cons_a(cons_a(z0))) -> c11(C(a(a(encArg(z0))))) ENCODE_B(cons_c(cons_c(z0))) -> c11(B(c(c(encArg(z0))))) ENCODE_B(cons_c(cons_b(z0))) -> c11(B(c(b(encArg(z0))))) ENCODE_B(cons_c(cons_a(z0))) -> c11(B(c(a(encArg(z0))))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) B(a(a(a(a(a(a(c(z0)))))))) -> c9(A(a(a(a(a(c(b(a(z0)))))))), A(a(b(a(a(a(c(z0))))))), A(b(a(a(a(c(z0)))))), B(a(a(a(c(z0)))))) B(a(a(a(a(a(a(a(a(a(z0)))))))))) -> c9(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(a(a(z0))))))))), A(b(a(a(a(a(a(a(z0)))))))), B(a(a(a(a(a(a(z0)))))))) B(a(a(a(a(a(a(x0))))))) -> c9(A(a(b(a(a(a(x0)))))), A(b(a(a(a(x0)))))) B(a(a(a(c(a(z0)))))) -> c9(A(a(a(a(b(c(z0)))))), A(a(b(c(a(z0))))), A(b(c(a(z0))))) B(a(a(a(c(a(a(z0))))))) -> c9(A(a(c(a(a(a(b(z0))))))), A(a(b(c(a(a(z0)))))), A(b(c(a(a(z0)))))) B(a(a(a(c(b(c(z0))))))) -> c9(A(a(c(b(c(b(a(z0))))))), A(a(b(c(b(c(z0)))))), A(b(c(b(c(z0)))))) B(a(a(a(c(x0))))) -> c9(A(a(b(c(x0)))), A(b(c(x0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, C_1, B_1, ENCODE_C_1 Compound Symbols: c10_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_1, c7_3, c7_2, c8_4, c8_3, c8_2, c8_1, c9_4, c9_2, c9_3 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_B(cons_b(z0)) -> c11(B(b(encArg(z0)))) by ENCODE_B(cons_b(cons_c(z0))) -> c11(B(b(c(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c11(B(b(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c11(B(b(a(encArg(z0))))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_c(z0)) -> c(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) c(b(a(a(z0)))) -> a(a(b(c(z0)))) c(c(b(b(z0)))) -> b(b(c(c(z0)))) b(a(a(a(z0)))) -> a(a(a(b(z0)))) a(b(c(z0))) -> c(b(a(z0))) Tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) ENCODE_B(cons_a(z0)) -> c11(B(a(encArg(z0)))) ENCODE_A(cons_c(z0)) -> c11(A(c(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c11(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c11(A(a(encArg(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c1(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_c(cons_c(cons_b(z0)))) -> c1(C(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c1(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_b(cons_c(z0)))) -> c1(C(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_c(cons_b(cons_b(z0)))) -> c1(C(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_c(cons_b(cons_a(z0)))) -> c1(C(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c1(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(cons_b(z0)))) -> c1(C(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c1(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_c(cons_c(z0)))) -> c2(B(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_b(cons_c(cons_b(z0)))) -> c2(B(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_b(cons_c(cons_a(z0)))) -> c2(B(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_c(z0)))) -> c2(B(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c2(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c2(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_c(z0)))) -> c2(B(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c2(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c2(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_c(cons_c(z0)))) -> c3(A(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ENCARG(cons_a(cons_c(cons_b(z0)))) -> c3(A(c(b(encArg(z0)))), ENCARG(cons_c(cons_b(z0)))) ENCARG(cons_a(cons_c(cons_a(z0)))) -> c3(A(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_c(z0)))) -> c3(A(b(c(encArg(z0)))), ENCARG(cons_b(cons_c(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c3(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c3(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_c(z0)))) -> c3(A(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c3(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c3(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) B(a(a(a(a(a(a(c(z0)))))))) -> c9(A(a(a(a(a(c(b(a(z0)))))))), A(a(b(a(a(a(c(z0))))))), A(b(a(a(a(c(z0)))))), B(a(a(a(c(z0)))))) B(a(a(a(a(a(a(a(a(a(z0)))))))))) -> c9(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(a(a(z0))))))))), A(b(a(a(a(a(a(a(z0)))))))), B(a(a(a(a(a(a(z0)))))))) B(a(a(a(a(a(a(x0))))))) -> c9(A(a(b(a(a(a(x0)))))), A(b(a(a(a(x0)))))) B(a(a(a(c(a(z0)))))) -> c9(A(a(a(a(b(c(z0)))))), A(a(b(c(a(z0))))), A(b(c(a(z0))))) B(a(a(a(c(a(a(z0))))))) -> c9(A(a(c(a(a(a(b(z0))))))), A(a(b(c(a(a(z0)))))), A(b(c(a(a(z0)))))) B(a(a(a(c(b(c(z0))))))) -> c9(A(a(c(b(c(b(a(z0))))))), A(a(b(c(b(c(z0)))))), A(b(c(b(c(z0)))))) B(a(a(a(c(x0))))) -> c9(A(a(b(c(x0)))), A(b(c(x0)))) ENCODE_C(cons_c(cons_c(z0))) -> c11(C(c(c(encArg(z0))))) ENCODE_C(cons_c(cons_b(z0))) -> c11(C(c(b(encArg(z0))))) ENCODE_C(cons_c(cons_a(z0))) -> c11(C(c(a(encArg(z0))))) ENCODE_C(cons_b(cons_c(z0))) -> c11(C(b(c(encArg(z0))))) ENCODE_C(cons_b(cons_b(z0))) -> c11(C(b(b(encArg(z0))))) ENCODE_C(cons_b(cons_a(z0))) -> c11(C(b(a(encArg(z0))))) ENCODE_C(cons_a(cons_c(z0))) -> c11(C(a(c(encArg(z0))))) ENCODE_C(cons_a(cons_b(z0))) -> c11(C(a(b(encArg(z0))))) ENCODE_C(cons_a(cons_a(z0))) -> c11(C(a(a(encArg(z0))))) ENCODE_B(cons_c(cons_c(z0))) -> c11(B(c(c(encArg(z0))))) ENCODE_B(cons_c(cons_b(z0))) -> c11(B(c(b(encArg(z0))))) ENCODE_B(cons_c(cons_a(z0))) -> c11(B(c(a(encArg(z0))))) ENCODE_B(cons_b(cons_c(z0))) -> c11(B(b(c(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c11(B(b(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c11(B(b(a(encArg(z0))))) S tuples: A(b(c(z0))) -> c10(C(b(a(z0))), B(a(z0)), A(z0)) C(b(a(a(b(a(a(b(a(a(z0)))))))))) -> c7(A(a(b(a(a(b(a(a(b(c(z0)))))))))), A(b(c(b(a(a(b(a(a(z0))))))))), B(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(b(a(a(b(a(a(c(b(b(z0)))))))))) -> c7(A(a(b(a(a(b(b(b(c(c(z0)))))))))), A(b(c(b(a(a(c(b(b(z0))))))))), B(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(b(a(a(b(a(a(x0))))))) -> c7(A(b(c(b(a(a(x0))))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(c(b(a(a(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(b(b(z0))))))))) -> c7(A(a(b(b(b(b(b(c(c(z0))))))))), A(b(c(c(b(b(b(b(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(b(a(a(c(b(b(b(a(a(z0)))))))))) -> c7(A(a(b(b(b(c(a(a(b(c(z0)))))))))), A(b(c(c(b(b(b(a(a(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(b(a(a(c(b(b(c(b(b(z0)))))))))) -> c7(A(a(b(b(b(c(b(b(c(c(z0)))))))))), A(b(c(c(b(b(c(b(b(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(b(a(a(c(b(b(x0))))))) -> c7(A(b(c(c(b(b(x0)))))), B(c(c(b(b(x0)))))) C(b(a(a(b(a(a(z0))))))) -> c7(A(b(c(b(a(a(z0)))))), B(a(a(b(c(z0))))), C(b(a(a(z0))))) C(b(a(a(c(b(b(z0))))))) -> c7(A(b(c(c(b(b(z0)))))), B(b(b(c(c(z0))))), C(c(b(b(z0))))) C(b(a(a(x0)))) -> c7(A(b(c(x0)))) C(c(b(b(b(a(a(b(a(a(z0)))))))))) -> c8(B(b(c(a(a(b(a(a(b(c(z0)))))))))), B(c(c(b(a(a(b(a(a(z0))))))))), C(c(b(a(a(b(a(a(z0)))))))), C(b(a(a(b(a(a(z0)))))))) C(c(b(b(b(a(a(c(b(b(z0)))))))))) -> c8(B(b(c(a(a(b(b(b(c(c(z0)))))))))), B(c(c(b(a(a(c(b(b(z0))))))))), C(c(b(a(a(c(b(b(z0)))))))), C(b(a(a(c(b(b(z0)))))))) C(c(b(b(b(a(a(z0))))))) -> c8(B(c(c(b(a(a(z0)))))), C(c(b(a(a(z0))))), C(b(a(a(z0))))) C(c(b(b(c(b(b(b(b(z0))))))))) -> c8(B(b(c(b(b(b(b(c(c(z0))))))))), B(c(c(c(b(b(b(b(z0)))))))), C(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(c(b(b(b(a(a(z0)))))))))) -> c8(B(b(c(b(b(c(a(a(b(c(z0)))))))))), B(c(c(c(b(b(b(a(a(z0))))))))), C(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(c(b(b(c(b(b(z0)))))))))) -> c8(B(b(c(b(b(c(b(b(c(c(z0)))))))))), B(c(c(c(b(b(c(b(b(z0))))))))), C(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(c(b(b(x0))))))) -> c8(B(c(c(c(b(b(x0)))))), C(c(c(b(b(x0)))))) C(c(b(b(b(b(b(b(z0)))))))) -> c8(B(b(b(b(b(b(c(c(z0)))))))), B(c(c(b(b(b(b(z0))))))), C(c(b(b(b(b(z0))))))) C(c(b(b(b(b(b(a(a(z0))))))))) -> c8(B(b(b(b(c(a(a(b(c(z0))))))))), B(c(c(b(b(b(a(a(z0)))))))), C(c(b(b(b(a(a(z0)))))))) C(c(b(b(b(b(c(b(b(z0))))))))) -> c8(B(b(b(b(c(b(b(c(c(z0))))))))), B(c(c(b(b(c(b(b(z0)))))))), C(c(b(b(c(b(b(z0)))))))) C(c(b(b(b(b(x0)))))) -> c8(B(c(c(b(b(x0)))))) B(a(a(a(a(a(a(c(z0)))))))) -> c9(A(a(a(a(a(c(b(a(z0)))))))), A(a(b(a(a(a(c(z0))))))), A(b(a(a(a(c(z0)))))), B(a(a(a(c(z0)))))) B(a(a(a(a(a(a(a(a(a(z0)))))))))) -> c9(A(a(a(a(a(a(a(a(a(b(z0)))))))))), A(a(b(a(a(a(a(a(a(z0))))))))), A(b(a(a(a(a(a(a(z0)))))))), B(a(a(a(a(a(a(z0)))))))) B(a(a(a(a(a(a(x0))))))) -> c9(A(a(b(a(a(a(x0)))))), A(b(a(a(a(x0)))))) B(a(a(a(c(a(z0)))))) -> c9(A(a(a(a(b(c(z0)))))), A(a(b(c(a(z0))))), A(b(c(a(z0))))) B(a(a(a(c(a(a(z0))))))) -> c9(A(a(c(a(a(a(b(z0))))))), A(a(b(c(a(a(z0)))))), A(b(c(a(a(z0)))))) B(a(a(a(c(b(c(z0))))))) -> c9(A(a(c(b(c(b(a(z0))))))), A(a(b(c(b(c(z0)))))), A(b(c(b(c(z0)))))) B(a(a(a(c(x0))))) -> c9(A(a(b(c(x0)))), A(b(c(x0)))) K tuples:none Defined Rule Symbols: encArg_1, c_1, b_1, a_1 Defined Pair Symbols: A_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, C_1, B_1, ENCODE_C_1 Compound Symbols: c10_3, c11_1, c1_2, c2_2, c3_2, c7_4, c7_1, c7_3, c7_2, c8_4, c8_3, c8_2, c8_1, c9_4, c9_2, c9_3