/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- KILLED proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 31 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 4 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 430 ms] (12) BOUNDS(1, INF) (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (14) TRS for Loop Detection (15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxTRS (17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (18) CpxRelTRS (19) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxWeightedTrs (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxTypedWeightedTrs (23) CompletionProof [UPPER BOUND(ID), 0 ms] (24) CpxTypedWeightedCompleteTrs (25) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (38) CdtProblem (39) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (42) CdtProblem (43) CdtUsableRulesProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem ---------------------------------------- (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: b(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(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(a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(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: b(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) The (relative) TRS S consists of the following rules: encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(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: b(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) The (relative) TRS S consists of the following rules: encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: b(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) The (relative) TRS S consists of the following rules: encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: b(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Types: b :: a:cons_b -> a:cons_b a :: a:cons_b -> a:cons_b encArg :: a:cons_b -> a:cons_b cons_b :: a:cons_b -> a:cons_b encode_b :: a:cons_b -> a:cons_b encode_a :: a:cons_b -> a:cons_b hole_a:cons_b1_0 :: a:cons_b gen_a:cons_b2_0 :: Nat -> a:cons_b ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: b, encArg They will be analysed ascendingly in the following order: b < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: b(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Types: b :: a:cons_b -> a:cons_b a :: a:cons_b -> a:cons_b encArg :: a:cons_b -> a:cons_b cons_b :: a:cons_b -> a:cons_b encode_b :: a:cons_b -> a:cons_b encode_a :: a:cons_b -> a:cons_b hole_a:cons_b1_0 :: a:cons_b gen_a:cons_b2_0 :: Nat -> a:cons_b Generator Equations: gen_a:cons_b2_0(0) <=> hole_a:cons_b1_0 gen_a:cons_b2_0(+(x, 1)) <=> a(gen_a:cons_b2_0(x)) The following defined symbols remain to be analysed: b, encArg They will be analysed ascendingly in the following order: b < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_a:cons_b2_0(+(1, n10_0))) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_a:cons_b2_0(+(1, 0))) Induction Step: encArg(gen_a:cons_b2_0(+(1, +(n10_0, 1)))) ->_R^Omega(0) a(encArg(gen_a:cons_b2_0(+(1, n10_0)))) ->_IH a(*3_0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (12) BOUNDS(1, INF) ---------------------------------------- (13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (14) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: b(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) The (relative) TRS S consists of the following rules: encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (15) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (16) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: b(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(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: b(a(a(c_b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) The (relative) TRS S consists of the following rules: encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) b(x0) -> c_b(x0) Rewrite Strategy: INNERMOST ---------------------------------------- (19) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (20) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: b(a(a(c_b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) [1] encArg(a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encode_b(x_1) -> b(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] b(x0) -> c_b(x0) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (21) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (22) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: b(a(a(c_b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) [1] encArg(a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encode_b(x_1) -> b(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] b(x0) -> c_b(x0) [0] The TRS has the following type information: b :: a:c_b:cons_b -> a:c_b:cons_b a :: a:c_b:cons_b -> a:c_b:cons_b c_b :: a:c_b:cons_b -> a:c_b:cons_b encArg :: a:c_b:cons_b -> a:c_b:cons_b cons_b :: a:c_b:cons_b -> a:c_b:cons_b encode_b :: a:c_b:cons_b -> a:c_b:cons_b encode_a :: a:c_b:cons_b -> a:c_b:cons_b Rewrite Strategy: INNERMOST ---------------------------------------- (23) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: encArg_1 encode_b_1 encode_a_1 b_1 Due to the following rules being added: encArg(v0) -> const [0] encode_b(v0) -> const [0] encode_a(v0) -> const [0] b(v0) -> const [0] And the following fresh constants: const ---------------------------------------- (24) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: b(a(a(c_b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) [1] encArg(a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encode_b(x_1) -> b(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] b(x0) -> c_b(x0) [0] encArg(v0) -> const [0] encode_b(v0) -> const [0] encode_a(v0) -> const [0] b(v0) -> const [0] The TRS has the following type information: b :: a:c_b:cons_b:const -> a:c_b:cons_b:const a :: a:c_b:cons_b:const -> a:c_b:cons_b:const c_b :: a:c_b:cons_b:const -> a:c_b:cons_b:const encArg :: a:c_b:cons_b:const -> a:c_b:cons_b:const cons_b :: a:c_b:cons_b:const -> a:c_b:cons_b:const encode_b :: a:c_b:cons_b:const -> a:c_b:cons_b:const encode_a :: a:c_b:cons_b:const -> a:c_b:cons_b:const const :: a:c_b:cons_b:const Rewrite Strategy: INNERMOST ---------------------------------------- (25) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (26) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: b(a(a(c_b(a(a(a(c_b(a(x1'))))))))) -> a(a(b(b(a(a(a(a(b(b(a(a(b(x1'))))))))))))) [2] b(a(a(c_b(a(x1))))) -> a(a(b(b(a(a(c_b(x1))))))) [1] b(a(a(c_b(a(x1))))) -> a(a(b(b(a(a(const)))))) [1] encArg(a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_b(a(x_1'))) -> b(a(encArg(x_1'))) [0] encArg(cons_b(cons_b(x_1''))) -> b(b(encArg(x_1''))) [0] encArg(cons_b(x_1)) -> b(const) [0] encode_b(a(x_11)) -> b(a(encArg(x_11))) [0] encode_b(cons_b(x_12)) -> b(b(encArg(x_12))) [0] encode_b(x_1) -> b(const) [0] encode_a(x_1) -> a(encArg(x_1)) [0] b(x0) -> c_b(x0) [0] encArg(v0) -> const [0] encode_b(v0) -> const [0] encode_a(v0) -> const [0] b(v0) -> const [0] The TRS has the following type information: b :: a:c_b:cons_b:const -> a:c_b:cons_b:const a :: a:c_b:cons_b:const -> a:c_b:cons_b:const c_b :: a:c_b:cons_b:const -> a:c_b:cons_b:const encArg :: a:c_b:cons_b:const -> a:c_b:cons_b:const cons_b :: a:c_b:cons_b:const -> a:c_b:cons_b:const encode_b :: a:c_b:cons_b:const -> a:c_b:cons_b:const encode_a :: a:c_b:cons_b:const -> a:c_b:cons_b:const const :: a:c_b:cons_b:const Rewrite Strategy: INNERMOST ---------------------------------------- (27) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: const => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 b(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 b(z) -{ 1 }-> 1 + (1 + b(b(1 + (1 + 0)))) :|: x1 >= 0, z = 1 + (1 + (1 + (1 + x1))) b(z) -{ 1 }-> 1 + (1 + b(b(1 + (1 + (1 + x1))))) :|: x1 >= 0, z = 1 + (1 + (1 + (1 + x1))) b(z) -{ 2 }-> 1 + (1 + b(b(1 + (1 + (1 + (1 + b(b(1 + (1 + b(x1')))))))))) :|: x1' >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + x1'))))))) encArg(z) -{ 0 }-> b(b(encArg(x_1''))) :|: z = 1 + (1 + x_1''), x_1'' >= 0 encArg(z) -{ 0 }-> b(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> b(1 + encArg(x_1')) :|: z = 1 + (1 + x_1'), x_1' >= 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0 encode_a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_a(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1 encode_b(z) -{ 0 }-> b(b(encArg(x_12))) :|: z = 1 + x_12, x_12 >= 0 encode_b(z) -{ 0 }-> b(0) :|: x_1 >= 0, z = x_1 encode_b(z) -{ 0 }-> b(1 + encArg(x_11)) :|: x_11 >= 0, z = 1 + x_11 encode_b(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: b(z) -{ 0 }-> 0 :|: z >= 0 b(z) -{ 0 }-> 1 + z :|: z >= 0 b(z) -{ 1 }-> 1 + (1 + b(b(1 + (1 + 0)))) :|: z - 4 >= 0 b(z) -{ 1 }-> 1 + (1 + b(b(1 + (1 + (1 + (z - 4)))))) :|: z - 4 >= 0 b(z) -{ 2 }-> 1 + (1 + b(b(1 + (1 + (1 + (1 + b(b(1 + (1 + b(z - 8)))))))))) :|: z - 8 >= 0 encArg(z) -{ 0 }-> b(b(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> b(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> b(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_a(z) -{ 0 }-> 0 :|: z >= 0 encode_a(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_b(z) -{ 0 }-> b(b(encArg(z - 1))) :|: z - 1 >= 0 encode_b(z) -{ 0 }-> b(0) :|: z >= 0 encode_b(z) -{ 0 }-> b(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_b(z) -{ 0 }-> 0 :|: z >= 0 ---------------------------------------- (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_b(v0) -> null_encode_b [0] encode_a(v0) -> null_encode_a [0] b(v0) -> null_b [0] And the following fresh constants: null_encArg, null_encode_b, null_encode_a, null_b ---------------------------------------- (32) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: b(a(a(c_b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) [1] encArg(a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encode_b(x_1) -> b(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] b(x0) -> c_b(x0) [0] encArg(v0) -> null_encArg [0] encode_b(v0) -> null_encode_b [0] encode_a(v0) -> null_encode_a [0] b(v0) -> null_b [0] The TRS has the following type information: b :: a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b -> a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b a :: a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b -> a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b c_b :: a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b -> a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b encArg :: a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b -> a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b cons_b :: a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b -> a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b encode_b :: a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b -> a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b encode_a :: a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b -> a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b null_encArg :: a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b null_encode_b :: a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b null_encode_a :: a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b null_b :: a:c_b:cons_b:null_encArg:null_encode_b:null_encode_a:null_b Rewrite Strategy: INNERMOST ---------------------------------------- (33) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: null_encArg => 0 null_encode_b => 0 null_encode_a => 0 null_b => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 b(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 b(z) -{ 1 }-> 1 + (1 + b(b(1 + (1 + b(x1))))) :|: x1 >= 0, z = 1 + (1 + (1 + (1 + x1))) encArg(z) -{ 0 }-> b(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0 encode_a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_a(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1 encode_b(z) -{ 0 }-> b(encArg(x_1)) :|: x_1 >= 0, z = x_1 encode_b(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(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) ENCARG(cons_b(z0)) -> c1(B(encArg(z0)), ENCARG(z0)) ENCODE_B(z0) -> c2(B(encArg(z0)), ENCARG(z0)) ENCODE_A(z0) -> c3(ENCARG(z0)) B(a(a(b(a(z0))))) -> c4(B(b(a(a(b(z0))))), B(a(a(b(z0)))), B(z0)) S tuples: B(a(a(b(a(z0))))) -> c4(B(b(a(a(b(z0))))), B(a(a(b(z0)))), B(z0)) K tuples:none Defined Rule Symbols: b_1, encArg_1, encode_b_1, encode_a_1 Defined Pair Symbols: ENCARG_1, ENCODE_B_1, ENCODE_A_1, B_1 Compound Symbols: c_1, c1_2, c2_2, c3_1, c4_3 ---------------------------------------- (37) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ENCODE_A(z0) -> c3(ENCARG(z0)) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) ENCARG(cons_b(z0)) -> c1(B(encArg(z0)), ENCARG(z0)) ENCODE_B(z0) -> c2(B(encArg(z0)), ENCARG(z0)) B(a(a(b(a(z0))))) -> c4(B(b(a(a(b(z0))))), B(a(a(b(z0)))), B(z0)) S tuples: B(a(a(b(a(z0))))) -> c4(B(b(a(a(b(z0))))), B(a(a(b(z0)))), B(z0)) K tuples:none Defined Rule Symbols: b_1, encArg_1, encode_b_1, encode_a_1 Defined Pair Symbols: ENCARG_1, ENCODE_B_1, B_1 Compound Symbols: c_1, c1_2, c2_2, c4_3 ---------------------------------------- (39) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) ENCARG(cons_b(z0)) -> c1(B(encArg(z0)), ENCARG(z0)) B(a(a(b(a(z0))))) -> c4(B(b(a(a(b(z0))))), B(a(a(b(z0)))), B(z0)) ENCODE_B(z0) -> c3(B(encArg(z0))) ENCODE_B(z0) -> c3(ENCARG(z0)) S tuples: B(a(a(b(a(z0))))) -> c4(B(b(a(a(b(z0))))), B(a(a(b(z0)))), B(z0)) K tuples:none Defined Rule Symbols: b_1, encArg_1, encode_b_1, encode_a_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c_1, c1_2, c4_3, c3_1 ---------------------------------------- (41) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ENCODE_B(z0) -> c3(ENCARG(z0)) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) ENCARG(cons_b(z0)) -> c1(B(encArg(z0)), ENCARG(z0)) B(a(a(b(a(z0))))) -> c4(B(b(a(a(b(z0))))), B(a(a(b(z0)))), B(z0)) ENCODE_B(z0) -> c3(B(encArg(z0))) S tuples: B(a(a(b(a(z0))))) -> c4(B(b(a(a(b(z0))))), B(a(a(b(z0)))), B(z0)) K tuples:none Defined Rule Symbols: b_1, encArg_1, encode_b_1, encode_a_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c_1, c1_2, c4_3, c3_1 ---------------------------------------- (43) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_b(z0) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) ENCARG(cons_b(z0)) -> c1(B(encArg(z0)), ENCARG(z0)) B(a(a(b(a(z0))))) -> c4(B(b(a(a(b(z0))))), B(a(a(b(z0)))), B(z0)) ENCODE_B(z0) -> c3(B(encArg(z0))) S tuples: B(a(a(b(a(z0))))) -> c4(B(b(a(a(b(z0))))), B(a(a(b(z0)))), B(z0)) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c_1, c1_2, c4_3, c3_1 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(z0)) -> c1(B(encArg(z0)), ENCARG(z0)) by ENCARG(cons_b(a(z0))) -> c1(B(a(encArg(z0))), ENCARG(a(z0))) ENCARG(cons_b(cons_b(z0))) -> c1(B(b(encArg(z0))), ENCARG(cons_b(z0))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) B(a(a(b(a(z0))))) -> c4(B(b(a(a(b(z0))))), B(a(a(b(z0)))), B(z0)) ENCODE_B(z0) -> c3(B(encArg(z0))) ENCARG(cons_b(a(z0))) -> c1(B(a(encArg(z0))), ENCARG(a(z0))) ENCARG(cons_b(cons_b(z0))) -> c1(B(b(encArg(z0))), ENCARG(cons_b(z0))) S tuples: B(a(a(b(a(z0))))) -> c4(B(b(a(a(b(z0))))), B(a(a(b(z0)))), B(z0)) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c_1, c4_3, c3_1, c1_2 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(a(a(b(a(z0))))) -> c4(B(b(a(a(b(z0))))), B(a(a(b(z0)))), B(z0)) by B(a(a(b(a(a(z0)))))) -> c4(B(a(a(b(b(a(a(b(z0)))))))), B(a(a(b(a(z0))))), B(a(z0))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(b(a(a(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) ENCODE_B(z0) -> c3(B(encArg(z0))) ENCARG(cons_b(a(z0))) -> c1(B(a(encArg(z0))), ENCARG(a(z0))) ENCARG(cons_b(cons_b(z0))) -> c1(B(b(encArg(z0))), ENCARG(cons_b(z0))) B(a(a(b(a(a(z0)))))) -> c4(B(a(a(b(b(a(a(b(z0)))))))), B(a(a(b(a(z0))))), B(a(z0))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(b(a(a(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) S tuples: B(a(a(b(a(a(z0)))))) -> c4(B(a(a(b(b(a(a(b(z0)))))))), B(a(a(b(a(z0))))), B(a(z0))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(b(a(a(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, ENCODE_B_1, B_1 Compound Symbols: c_1, c3_1, c1_2, c4_3 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) ENCODE_B(z0) -> c3(B(encArg(z0))) ENCARG(cons_b(a(z0))) -> c1(B(a(encArg(z0))), ENCARG(a(z0))) ENCARG(cons_b(cons_b(z0))) -> c1(B(b(encArg(z0))), ENCARG(cons_b(z0))) B(a(a(b(a(a(z0)))))) -> c4(B(a(a(b(b(a(a(b(z0)))))))), B(a(a(b(a(z0))))), B(a(z0))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) S tuples: B(a(a(b(a(a(z0)))))) -> c4(B(a(a(b(b(a(a(b(z0)))))))), B(a(a(b(a(z0))))), B(a(z0))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, ENCODE_B_1, B_1 Compound Symbols: c_1, c3_1, c1_2, c4_3, c4_2 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_B(z0) -> c3(B(encArg(z0))) by ENCODE_B(a(z0)) -> c3(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c3(B(b(encArg(z0)))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) ENCARG(cons_b(a(z0))) -> c1(B(a(encArg(z0))), ENCARG(a(z0))) ENCARG(cons_b(cons_b(z0))) -> c1(B(b(encArg(z0))), ENCARG(cons_b(z0))) B(a(a(b(a(a(z0)))))) -> c4(B(a(a(b(b(a(a(b(z0)))))))), B(a(a(b(a(z0))))), B(a(z0))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) ENCODE_B(a(z0)) -> c3(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c3(B(b(encArg(z0)))) S tuples: B(a(a(b(a(a(z0)))))) -> c4(B(a(a(b(b(a(a(b(z0)))))))), B(a(a(b(a(z0))))), B(a(z0))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c_1, c1_2, c4_3, c4_2, c3_1 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(a(z0))) -> c1(B(a(encArg(z0))), ENCARG(a(z0))) by ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(a(x0))) -> c1(ENCARG(a(x0))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) ENCARG(cons_b(cons_b(z0))) -> c1(B(b(encArg(z0))), ENCARG(cons_b(z0))) B(a(a(b(a(a(z0)))))) -> c4(B(a(a(b(b(a(a(b(z0)))))))), B(a(a(b(a(z0))))), B(a(z0))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) ENCODE_B(a(z0)) -> c3(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c3(B(b(encArg(z0)))) ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(a(x0))) -> c1(ENCARG(a(x0))) S tuples: B(a(a(b(a(a(z0)))))) -> c4(B(a(a(b(b(a(a(b(z0)))))))), B(a(a(b(a(z0))))), B(a(z0))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c_1, c1_2, c4_3, c4_2, c3_1, c1_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(cons_b(z0))) -> c1(B(b(encArg(z0))), ENCARG(cons_b(z0))) by ENCARG(cons_b(cons_b(a(z0)))) -> c1(B(b(a(encArg(z0)))), ENCARG(cons_b(a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c1(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) B(a(a(b(a(a(z0)))))) -> c4(B(a(a(b(b(a(a(b(z0)))))))), B(a(a(b(a(z0))))), B(a(z0))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) ENCODE_B(a(z0)) -> c3(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c3(B(b(encArg(z0)))) ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(a(x0))) -> c1(ENCARG(a(x0))) ENCARG(cons_b(cons_b(a(z0)))) -> c1(B(b(a(encArg(z0)))), ENCARG(cons_b(a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c1(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) S tuples: B(a(a(b(a(a(z0)))))) -> c4(B(a(a(b(b(a(a(b(z0)))))))), B(a(a(b(a(z0))))), B(a(z0))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c_1, c4_3, c4_2, c3_1, c1_2, c1_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(a(a(b(a(a(z0)))))) -> c4(B(a(a(b(b(a(a(b(z0)))))))), B(a(a(b(a(z0))))), B(a(z0))) by B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(b(a(a(a(a(b(b(a(a(b(z0)))))))))))))), B(a(a(b(a(a(a(b(a(z0))))))))), B(a(a(a(b(a(z0))))))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) ENCODE_B(a(z0)) -> c3(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c3(B(b(encArg(z0)))) ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(a(x0))) -> c1(ENCARG(a(x0))) ENCARG(cons_b(cons_b(a(z0)))) -> c1(B(b(a(encArg(z0)))), ENCARG(cons_b(a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c1(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(b(a(a(a(a(b(b(a(a(b(z0)))))))))))))), B(a(a(b(a(a(a(b(a(z0))))))))), B(a(a(a(b(a(z0))))))) S tuples: B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(b(a(a(a(a(b(b(a(a(b(z0)))))))))))))), B(a(a(b(a(a(a(b(a(z0))))))))), B(a(a(a(b(a(z0))))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c_1, c4_2, c3_1, c1_2, c1_1, c4_3 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) ENCODE_B(a(z0)) -> c3(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c3(B(b(encArg(z0)))) ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(a(x0))) -> c1(ENCARG(a(x0))) ENCARG(cons_b(cons_b(a(z0)))) -> c1(B(b(a(encArg(z0)))), ENCARG(cons_b(a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c1(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) S tuples: B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c_1, c4_2, c3_1, c1_2, c1_1, c4_3, c4_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(a(b(a(z0)))))))), B(a(a(b(a(z0)))))) by B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(a(a(b(b(a(a(b(z0)))))))))), B(a(a(b(a(z0)))))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) ENCODE_B(a(z0)) -> c3(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c3(B(b(encArg(z0)))) ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(a(x0))) -> c1(ENCARG(a(x0))) ENCARG(cons_b(cons_b(a(z0)))) -> c1(B(b(a(encArg(z0)))), ENCARG(cons_b(a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c1(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(a(a(b(b(a(a(b(z0)))))))))), B(a(a(b(a(z0)))))) S tuples: B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(a(a(b(b(a(a(b(z0)))))))))), B(a(a(b(a(z0)))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, ENCODE_B_1, B_1 Compound Symbols: c_1, c3_1, c1_2, c1_1, c4_3, c4_1, c4_2 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) ENCODE_B(a(z0)) -> c3(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c3(B(b(encArg(z0)))) ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(a(x0))) -> c1(ENCARG(a(x0))) ENCARG(cons_b(cons_b(a(z0)))) -> c1(B(b(a(encArg(z0)))), ENCARG(cons_b(a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c1(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(z0)))))) S tuples: B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(z0)))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, ENCODE_B_1, B_1 Compound Symbols: c_1, c3_1, c1_2, c1_1, c4_3, c4_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_B(a(z0)) -> c3(B(a(encArg(z0)))) by ENCODE_B(a(a(z0))) -> c3(B(a(a(encArg(z0))))) ENCODE_B(a(cons_b(z0))) -> c3(B(a(b(encArg(z0))))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) ENCODE_B(cons_b(z0)) -> c3(B(b(encArg(z0)))) ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(a(x0))) -> c1(ENCARG(a(x0))) ENCARG(cons_b(cons_b(a(z0)))) -> c1(B(b(a(encArg(z0)))), ENCARG(cons_b(a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c1(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(z0)))))) ENCODE_B(a(a(z0))) -> c3(B(a(a(encArg(z0))))) ENCODE_B(a(cons_b(z0))) -> c3(B(a(b(encArg(z0))))) S tuples: B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(z0)))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, ENCODE_B_1, B_1 Compound Symbols: c_1, c3_1, c1_2, c1_1, c4_3, c4_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_B(cons_b(z0)) -> c3(B(b(encArg(z0)))) by ENCODE_B(cons_b(a(z0))) -> c3(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c3(B(b(b(encArg(z0))))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(a(z0)) -> c(ENCARG(z0)) ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(a(x0))) -> c1(ENCARG(a(x0))) ENCARG(cons_b(cons_b(a(z0)))) -> c1(B(b(a(encArg(z0)))), ENCARG(cons_b(a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c1(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(z0)))))) ENCODE_B(a(a(z0))) -> c3(B(a(a(encArg(z0))))) ENCODE_B(a(cons_b(z0))) -> c3(B(a(b(encArg(z0))))) ENCODE_B(cons_b(a(z0))) -> c3(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c3(B(b(b(encArg(z0))))) S tuples: B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(z0)))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c_1, c1_2, c1_1, c4_3, c4_1, c3_1 ---------------------------------------- (69) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(a(z0)) -> c(ENCARG(z0)) by ENCARG(a(a(y0))) -> c(ENCARG(a(y0))) ENCARG(a(cons_b(a(a(y0))))) -> c(ENCARG(cons_b(a(a(y0))))) ENCARG(a(cons_b(a(cons_b(y0))))) -> c(ENCARG(cons_b(a(cons_b(y0))))) ENCARG(a(cons_b(a(y0)))) -> c(ENCARG(cons_b(a(y0)))) ENCARG(a(cons_b(cons_b(a(y0))))) -> c(ENCARG(cons_b(cons_b(a(y0))))) ENCARG(a(cons_b(cons_b(cons_b(y0))))) -> c(ENCARG(cons_b(cons_b(cons_b(y0))))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(a(x0))) -> c1(ENCARG(a(x0))) ENCARG(cons_b(cons_b(a(z0)))) -> c1(B(b(a(encArg(z0)))), ENCARG(cons_b(a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c1(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(z0)))))) ENCODE_B(a(a(z0))) -> c3(B(a(a(encArg(z0))))) ENCODE_B(a(cons_b(z0))) -> c3(B(a(b(encArg(z0))))) ENCODE_B(cons_b(a(z0))) -> c3(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c3(B(b(b(encArg(z0))))) ENCARG(a(a(y0))) -> c(ENCARG(a(y0))) ENCARG(a(cons_b(a(a(y0))))) -> c(ENCARG(cons_b(a(a(y0))))) ENCARG(a(cons_b(a(cons_b(y0))))) -> c(ENCARG(cons_b(a(cons_b(y0))))) ENCARG(a(cons_b(a(y0)))) -> c(ENCARG(cons_b(a(y0)))) ENCARG(a(cons_b(cons_b(a(y0))))) -> c(ENCARG(cons_b(cons_b(a(y0))))) ENCARG(a(cons_b(cons_b(cons_b(y0))))) -> c(ENCARG(cons_b(cons_b(cons_b(y0))))) S tuples: B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(z0)))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c1_2, c1_1, c4_3, c4_1, c3_1, c_1 ---------------------------------------- (71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_b(a(x0))) -> c1(ENCARG(a(x0))) by ENCARG(cons_b(a(a(y0)))) -> c1(ENCARG(a(a(y0)))) ENCARG(cons_b(a(cons_b(a(a(y0)))))) -> c1(ENCARG(a(cons_b(a(a(y0)))))) ENCARG(cons_b(a(cons_b(a(cons_b(y0)))))) -> c1(ENCARG(a(cons_b(a(cons_b(y0)))))) ENCARG(cons_b(a(cons_b(a(y0))))) -> c1(ENCARG(a(cons_b(a(y0))))) ENCARG(cons_b(a(cons_b(cons_b(a(y0)))))) -> c1(ENCARG(a(cons_b(cons_b(a(y0)))))) ENCARG(cons_b(a(cons_b(cons_b(cons_b(y0)))))) -> c1(ENCARG(a(cons_b(cons_b(cons_b(y0)))))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(cons_b(a(z0)))) -> c1(B(b(a(encArg(z0)))), ENCARG(cons_b(a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c1(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(z0)))))) ENCODE_B(a(a(z0))) -> c3(B(a(a(encArg(z0))))) ENCODE_B(a(cons_b(z0))) -> c3(B(a(b(encArg(z0))))) ENCODE_B(cons_b(a(z0))) -> c3(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c3(B(b(b(encArg(z0))))) ENCARG(a(a(y0))) -> c(ENCARG(a(y0))) ENCARG(a(cons_b(a(a(y0))))) -> c(ENCARG(cons_b(a(a(y0))))) ENCARG(a(cons_b(a(cons_b(y0))))) -> c(ENCARG(cons_b(a(cons_b(y0))))) ENCARG(a(cons_b(a(y0)))) -> c(ENCARG(cons_b(a(y0)))) ENCARG(a(cons_b(cons_b(a(y0))))) -> c(ENCARG(cons_b(cons_b(a(y0))))) ENCARG(a(cons_b(cons_b(cons_b(y0))))) -> c(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(cons_b(a(a(y0)))) -> c1(ENCARG(a(a(y0)))) ENCARG(cons_b(a(cons_b(a(a(y0)))))) -> c1(ENCARG(a(cons_b(a(a(y0)))))) ENCARG(cons_b(a(cons_b(a(cons_b(y0)))))) -> c1(ENCARG(a(cons_b(a(cons_b(y0)))))) ENCARG(cons_b(a(cons_b(a(y0))))) -> c1(ENCARG(a(cons_b(a(y0))))) ENCARG(cons_b(a(cons_b(cons_b(a(y0)))))) -> c1(ENCARG(a(cons_b(cons_b(a(y0)))))) ENCARG(cons_b(a(cons_b(cons_b(cons_b(y0)))))) -> c1(ENCARG(a(cons_b(cons_b(cons_b(y0)))))) S tuples: B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(z0)))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c1_2, c4_3, c4_1, c3_1, c_1, c1_1 ---------------------------------------- (73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace B(a(a(b(a(a(a(b(a(z0))))))))) -> c4(B(a(a(b(a(z0)))))) by B(a(a(b(a(a(a(b(a(a(a(y0))))))))))) -> c4(B(a(a(b(a(a(a(y0)))))))) B(a(a(b(a(a(a(b(a(a(a(a(b(a(y0)))))))))))))) -> c4(B(a(a(b(a(a(a(a(b(a(y0))))))))))) B(a(a(b(a(a(a(b(a(a(a(b(a(y0))))))))))))) -> c4(B(a(a(b(a(a(a(b(a(y0)))))))))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(cons_b(a(z0)))) -> c1(B(b(a(encArg(z0)))), ENCARG(cons_b(a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c1(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) ENCODE_B(a(a(z0))) -> c3(B(a(a(encArg(z0))))) ENCODE_B(a(cons_b(z0))) -> c3(B(a(b(encArg(z0))))) ENCODE_B(cons_b(a(z0))) -> c3(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c3(B(b(b(encArg(z0))))) ENCARG(a(a(y0))) -> c(ENCARG(a(y0))) ENCARG(a(cons_b(a(a(y0))))) -> c(ENCARG(cons_b(a(a(y0))))) ENCARG(a(cons_b(a(cons_b(y0))))) -> c(ENCARG(cons_b(a(cons_b(y0))))) ENCARG(a(cons_b(a(y0)))) -> c(ENCARG(cons_b(a(y0)))) ENCARG(a(cons_b(cons_b(a(y0))))) -> c(ENCARG(cons_b(cons_b(a(y0))))) ENCARG(a(cons_b(cons_b(cons_b(y0))))) -> c(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(cons_b(a(a(y0)))) -> c1(ENCARG(a(a(y0)))) ENCARG(cons_b(a(cons_b(a(a(y0)))))) -> c1(ENCARG(a(cons_b(a(a(y0)))))) ENCARG(cons_b(a(cons_b(a(cons_b(y0)))))) -> c1(ENCARG(a(cons_b(a(cons_b(y0)))))) ENCARG(cons_b(a(cons_b(a(y0))))) -> c1(ENCARG(a(cons_b(a(y0))))) ENCARG(cons_b(a(cons_b(cons_b(a(y0)))))) -> c1(ENCARG(a(cons_b(cons_b(a(y0)))))) ENCARG(cons_b(a(cons_b(cons_b(cons_b(y0)))))) -> c1(ENCARG(a(cons_b(cons_b(cons_b(y0)))))) B(a(a(b(a(a(a(b(a(a(a(y0))))))))))) -> c4(B(a(a(b(a(a(a(y0)))))))) B(a(a(b(a(a(a(b(a(a(a(a(b(a(y0)))))))))))))) -> c4(B(a(a(b(a(a(a(a(b(a(y0))))))))))) B(a(a(b(a(a(a(b(a(a(a(b(a(y0))))))))))))) -> c4(B(a(a(b(a(a(a(b(a(y0)))))))))) S tuples: B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(a(a(y0))))))))))) -> c4(B(a(a(b(a(a(a(y0)))))))) B(a(a(b(a(a(a(b(a(a(a(a(b(a(y0)))))))))))))) -> c4(B(a(a(b(a(a(a(a(b(a(y0))))))))))) B(a(a(b(a(a(a(b(a(a(a(b(a(y0))))))))))))) -> c4(B(a(a(b(a(a(a(b(a(y0)))))))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c1_2, c4_3, c4_1, c3_1, c_1, c1_1 ---------------------------------------- (75) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(a(a(y0))) -> c(ENCARG(a(y0))) by ENCARG(a(a(a(y0)))) -> c(ENCARG(a(a(y0)))) ENCARG(a(a(cons_b(a(a(y0)))))) -> c(ENCARG(a(cons_b(a(a(y0)))))) ENCARG(a(a(cons_b(a(cons_b(y0)))))) -> c(ENCARG(a(cons_b(a(cons_b(y0)))))) ENCARG(a(a(cons_b(a(y0))))) -> c(ENCARG(a(cons_b(a(y0))))) ENCARG(a(a(cons_b(cons_b(a(y0)))))) -> c(ENCARG(a(cons_b(cons_b(a(y0)))))) ENCARG(a(a(cons_b(cons_b(cons_b(y0)))))) -> c(ENCARG(a(cons_b(cons_b(cons_b(y0)))))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(cons_b(a(z0)))) -> c1(B(b(a(encArg(z0)))), ENCARG(cons_b(a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c1(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) ENCODE_B(a(a(z0))) -> c3(B(a(a(encArg(z0))))) ENCODE_B(a(cons_b(z0))) -> c3(B(a(b(encArg(z0))))) ENCODE_B(cons_b(a(z0))) -> c3(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c3(B(b(b(encArg(z0))))) ENCARG(a(cons_b(a(a(y0))))) -> c(ENCARG(cons_b(a(a(y0))))) ENCARG(a(cons_b(a(cons_b(y0))))) -> c(ENCARG(cons_b(a(cons_b(y0))))) ENCARG(a(cons_b(a(y0)))) -> c(ENCARG(cons_b(a(y0)))) ENCARG(a(cons_b(cons_b(a(y0))))) -> c(ENCARG(cons_b(cons_b(a(y0))))) ENCARG(a(cons_b(cons_b(cons_b(y0))))) -> c(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(cons_b(a(a(y0)))) -> c1(ENCARG(a(a(y0)))) ENCARG(cons_b(a(cons_b(a(a(y0)))))) -> c1(ENCARG(a(cons_b(a(a(y0)))))) ENCARG(cons_b(a(cons_b(a(cons_b(y0)))))) -> c1(ENCARG(a(cons_b(a(cons_b(y0)))))) ENCARG(cons_b(a(cons_b(a(y0))))) -> c1(ENCARG(a(cons_b(a(y0))))) ENCARG(cons_b(a(cons_b(cons_b(a(y0)))))) -> c1(ENCARG(a(cons_b(cons_b(a(y0)))))) ENCARG(cons_b(a(cons_b(cons_b(cons_b(y0)))))) -> c1(ENCARG(a(cons_b(cons_b(cons_b(y0)))))) B(a(a(b(a(a(a(b(a(a(a(y0))))))))))) -> c4(B(a(a(b(a(a(a(y0)))))))) B(a(a(b(a(a(a(b(a(a(a(a(b(a(y0)))))))))))))) -> c4(B(a(a(b(a(a(a(a(b(a(y0))))))))))) B(a(a(b(a(a(a(b(a(a(a(b(a(y0))))))))))))) -> c4(B(a(a(b(a(a(a(b(a(y0)))))))))) ENCARG(a(a(a(y0)))) -> c(ENCARG(a(a(y0)))) ENCARG(a(a(cons_b(a(a(y0)))))) -> c(ENCARG(a(cons_b(a(a(y0)))))) ENCARG(a(a(cons_b(a(cons_b(y0)))))) -> c(ENCARG(a(cons_b(a(cons_b(y0)))))) ENCARG(a(a(cons_b(a(y0))))) -> c(ENCARG(a(cons_b(a(y0))))) ENCARG(a(a(cons_b(cons_b(a(y0)))))) -> c(ENCARG(a(cons_b(cons_b(a(y0)))))) ENCARG(a(a(cons_b(cons_b(cons_b(y0)))))) -> c(ENCARG(a(cons_b(cons_b(cons_b(y0)))))) S tuples: B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(a(a(y0))))))))))) -> c4(B(a(a(b(a(a(a(y0)))))))) B(a(a(b(a(a(a(b(a(a(a(a(b(a(y0)))))))))))))) -> c4(B(a(a(b(a(a(a(a(b(a(y0))))))))))) B(a(a(b(a(a(a(b(a(a(a(b(a(y0))))))))))))) -> c4(B(a(a(b(a(a(a(b(a(y0)))))))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c1_2, c4_3, c4_1, c3_1, c_1, c1_1 ---------------------------------------- (77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(a(cons_b(a(y0)))) -> c(ENCARG(cons_b(a(y0)))) by ENCARG(a(cons_b(a(a(y0))))) -> c(ENCARG(cons_b(a(a(y0))))) ENCARG(a(cons_b(a(cons_b(y0))))) -> c(ENCARG(cons_b(a(cons_b(y0))))) ENCARG(a(cons_b(a(cons_b(a(a(y0))))))) -> c(ENCARG(cons_b(a(cons_b(a(a(y0))))))) ENCARG(a(cons_b(a(cons_b(a(cons_b(y0))))))) -> c(ENCARG(cons_b(a(cons_b(a(cons_b(y0))))))) ENCARG(a(cons_b(a(cons_b(a(y0)))))) -> c(ENCARG(cons_b(a(cons_b(a(y0)))))) ENCARG(a(cons_b(a(cons_b(cons_b(a(y0))))))) -> c(ENCARG(cons_b(a(cons_b(cons_b(a(y0))))))) ENCARG(a(cons_b(a(cons_b(cons_b(cons_b(y0))))))) -> c(ENCARG(cons_b(a(cons_b(cons_b(cons_b(y0))))))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(cons_b(a(z0)))) -> c1(B(b(a(encArg(z0)))), ENCARG(cons_b(a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c1(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) ENCODE_B(a(a(z0))) -> c3(B(a(a(encArg(z0))))) ENCODE_B(a(cons_b(z0))) -> c3(B(a(b(encArg(z0))))) ENCODE_B(cons_b(a(z0))) -> c3(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c3(B(b(b(encArg(z0))))) ENCARG(a(cons_b(a(a(y0))))) -> c(ENCARG(cons_b(a(a(y0))))) ENCARG(a(cons_b(a(cons_b(y0))))) -> c(ENCARG(cons_b(a(cons_b(y0))))) ENCARG(a(cons_b(cons_b(a(y0))))) -> c(ENCARG(cons_b(cons_b(a(y0))))) ENCARG(a(cons_b(cons_b(cons_b(y0))))) -> c(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(cons_b(a(a(y0)))) -> c1(ENCARG(a(a(y0)))) ENCARG(cons_b(a(cons_b(a(a(y0)))))) -> c1(ENCARG(a(cons_b(a(a(y0)))))) ENCARG(cons_b(a(cons_b(a(cons_b(y0)))))) -> c1(ENCARG(a(cons_b(a(cons_b(y0)))))) ENCARG(cons_b(a(cons_b(a(y0))))) -> c1(ENCARG(a(cons_b(a(y0))))) ENCARG(cons_b(a(cons_b(cons_b(a(y0)))))) -> c1(ENCARG(a(cons_b(cons_b(a(y0)))))) ENCARG(cons_b(a(cons_b(cons_b(cons_b(y0)))))) -> c1(ENCARG(a(cons_b(cons_b(cons_b(y0)))))) B(a(a(b(a(a(a(b(a(a(a(y0))))))))))) -> c4(B(a(a(b(a(a(a(y0)))))))) B(a(a(b(a(a(a(b(a(a(a(a(b(a(y0)))))))))))))) -> c4(B(a(a(b(a(a(a(a(b(a(y0))))))))))) B(a(a(b(a(a(a(b(a(a(a(b(a(y0))))))))))))) -> c4(B(a(a(b(a(a(a(b(a(y0)))))))))) ENCARG(a(a(a(y0)))) -> c(ENCARG(a(a(y0)))) ENCARG(a(a(cons_b(a(a(y0)))))) -> c(ENCARG(a(cons_b(a(a(y0)))))) ENCARG(a(a(cons_b(a(cons_b(y0)))))) -> c(ENCARG(a(cons_b(a(cons_b(y0)))))) ENCARG(a(a(cons_b(a(y0))))) -> c(ENCARG(a(cons_b(a(y0))))) ENCARG(a(a(cons_b(cons_b(a(y0)))))) -> c(ENCARG(a(cons_b(cons_b(a(y0)))))) ENCARG(a(a(cons_b(cons_b(cons_b(y0)))))) -> c(ENCARG(a(cons_b(cons_b(cons_b(y0)))))) ENCARG(a(cons_b(a(cons_b(a(a(y0))))))) -> c(ENCARG(cons_b(a(cons_b(a(a(y0))))))) ENCARG(a(cons_b(a(cons_b(a(cons_b(y0))))))) -> c(ENCARG(cons_b(a(cons_b(a(cons_b(y0))))))) ENCARG(a(cons_b(a(cons_b(a(y0)))))) -> c(ENCARG(cons_b(a(cons_b(a(y0)))))) ENCARG(a(cons_b(a(cons_b(cons_b(a(y0))))))) -> c(ENCARG(cons_b(a(cons_b(cons_b(a(y0))))))) ENCARG(a(cons_b(a(cons_b(cons_b(cons_b(y0))))))) -> c(ENCARG(cons_b(a(cons_b(cons_b(cons_b(y0))))))) S tuples: B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(a(a(y0))))))))))) -> c4(B(a(a(b(a(a(a(y0)))))))) B(a(a(b(a(a(a(b(a(a(a(a(b(a(y0)))))))))))))) -> c4(B(a(a(b(a(a(a(a(b(a(y0))))))))))) B(a(a(b(a(a(a(b(a(a(a(b(a(y0))))))))))))) -> c4(B(a(a(b(a(a(a(b(a(y0)))))))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c1_2, c4_3, c4_1, c3_1, c_1, c1_1 ---------------------------------------- (79) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_b(a(a(y0)))) -> c1(ENCARG(a(a(y0)))) by ENCARG(cons_b(a(a(a(y0))))) -> c1(ENCARG(a(a(a(y0))))) ENCARG(cons_b(a(a(cons_b(a(a(y0))))))) -> c1(ENCARG(a(a(cons_b(a(a(y0))))))) ENCARG(cons_b(a(a(cons_b(a(cons_b(y0))))))) -> c1(ENCARG(a(a(cons_b(a(cons_b(y0))))))) ENCARG(cons_b(a(a(cons_b(a(y0)))))) -> c1(ENCARG(a(a(cons_b(a(y0)))))) ENCARG(cons_b(a(a(cons_b(cons_b(a(y0))))))) -> c1(ENCARG(a(a(cons_b(cons_b(a(y0))))))) ENCARG(cons_b(a(a(cons_b(cons_b(cons_b(y0))))))) -> c1(ENCARG(a(a(cons_b(cons_b(cons_b(y0))))))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(cons_b(a(z0)))) -> c1(B(b(a(encArg(z0)))), ENCARG(cons_b(a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c1(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) ENCODE_B(a(a(z0))) -> c3(B(a(a(encArg(z0))))) ENCODE_B(a(cons_b(z0))) -> c3(B(a(b(encArg(z0))))) ENCODE_B(cons_b(a(z0))) -> c3(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c3(B(b(b(encArg(z0))))) ENCARG(a(cons_b(a(a(y0))))) -> c(ENCARG(cons_b(a(a(y0))))) ENCARG(a(cons_b(a(cons_b(y0))))) -> c(ENCARG(cons_b(a(cons_b(y0))))) ENCARG(a(cons_b(cons_b(a(y0))))) -> c(ENCARG(cons_b(cons_b(a(y0))))) ENCARG(a(cons_b(cons_b(cons_b(y0))))) -> c(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(cons_b(a(cons_b(a(a(y0)))))) -> c1(ENCARG(a(cons_b(a(a(y0)))))) ENCARG(cons_b(a(cons_b(a(cons_b(y0)))))) -> c1(ENCARG(a(cons_b(a(cons_b(y0)))))) ENCARG(cons_b(a(cons_b(a(y0))))) -> c1(ENCARG(a(cons_b(a(y0))))) ENCARG(cons_b(a(cons_b(cons_b(a(y0)))))) -> c1(ENCARG(a(cons_b(cons_b(a(y0)))))) ENCARG(cons_b(a(cons_b(cons_b(cons_b(y0)))))) -> c1(ENCARG(a(cons_b(cons_b(cons_b(y0)))))) B(a(a(b(a(a(a(b(a(a(a(y0))))))))))) -> c4(B(a(a(b(a(a(a(y0)))))))) B(a(a(b(a(a(a(b(a(a(a(a(b(a(y0)))))))))))))) -> c4(B(a(a(b(a(a(a(a(b(a(y0))))))))))) B(a(a(b(a(a(a(b(a(a(a(b(a(y0))))))))))))) -> c4(B(a(a(b(a(a(a(b(a(y0)))))))))) ENCARG(a(a(a(y0)))) -> c(ENCARG(a(a(y0)))) ENCARG(a(a(cons_b(a(a(y0)))))) -> c(ENCARG(a(cons_b(a(a(y0)))))) ENCARG(a(a(cons_b(a(cons_b(y0)))))) -> c(ENCARG(a(cons_b(a(cons_b(y0)))))) ENCARG(a(a(cons_b(a(y0))))) -> c(ENCARG(a(cons_b(a(y0))))) ENCARG(a(a(cons_b(cons_b(a(y0)))))) -> c(ENCARG(a(cons_b(cons_b(a(y0)))))) ENCARG(a(a(cons_b(cons_b(cons_b(y0)))))) -> c(ENCARG(a(cons_b(cons_b(cons_b(y0)))))) ENCARG(a(cons_b(a(cons_b(a(a(y0))))))) -> c(ENCARG(cons_b(a(cons_b(a(a(y0))))))) ENCARG(a(cons_b(a(cons_b(a(cons_b(y0))))))) -> c(ENCARG(cons_b(a(cons_b(a(cons_b(y0))))))) ENCARG(a(cons_b(a(cons_b(a(y0)))))) -> c(ENCARG(cons_b(a(cons_b(a(y0)))))) ENCARG(a(cons_b(a(cons_b(cons_b(a(y0))))))) -> c(ENCARG(cons_b(a(cons_b(cons_b(a(y0))))))) ENCARG(a(cons_b(a(cons_b(cons_b(cons_b(y0))))))) -> c(ENCARG(cons_b(a(cons_b(cons_b(cons_b(y0))))))) ENCARG(cons_b(a(a(a(y0))))) -> c1(ENCARG(a(a(a(y0))))) ENCARG(cons_b(a(a(cons_b(a(a(y0))))))) -> c1(ENCARG(a(a(cons_b(a(a(y0))))))) ENCARG(cons_b(a(a(cons_b(a(cons_b(y0))))))) -> c1(ENCARG(a(a(cons_b(a(cons_b(y0))))))) ENCARG(cons_b(a(a(cons_b(a(y0)))))) -> c1(ENCARG(a(a(cons_b(a(y0)))))) ENCARG(cons_b(a(a(cons_b(cons_b(a(y0))))))) -> c1(ENCARG(a(a(cons_b(cons_b(a(y0))))))) ENCARG(cons_b(a(a(cons_b(cons_b(cons_b(y0))))))) -> c1(ENCARG(a(a(cons_b(cons_b(cons_b(y0))))))) S tuples: B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(a(a(y0))))))))))) -> c4(B(a(a(b(a(a(a(y0)))))))) B(a(a(b(a(a(a(b(a(a(a(a(b(a(y0)))))))))))))) -> c4(B(a(a(b(a(a(a(a(b(a(y0))))))))))) B(a(a(b(a(a(a(b(a(a(a(b(a(y0))))))))))))) -> c4(B(a(a(b(a(a(a(b(a(y0)))))))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c1_2, c4_3, c4_1, c3_1, c_1, c1_1 ---------------------------------------- (81) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_b(a(cons_b(a(y0))))) -> c1(ENCARG(a(cons_b(a(y0))))) by ENCARG(cons_b(a(cons_b(a(a(y0)))))) -> c1(ENCARG(a(cons_b(a(a(y0)))))) ENCARG(cons_b(a(cons_b(a(cons_b(y0)))))) -> c1(ENCARG(a(cons_b(a(cons_b(y0)))))) ENCARG(cons_b(a(cons_b(a(cons_b(a(a(y0)))))))) -> c1(ENCARG(a(cons_b(a(cons_b(a(a(y0)))))))) ENCARG(cons_b(a(cons_b(a(cons_b(a(cons_b(y0)))))))) -> c1(ENCARG(a(cons_b(a(cons_b(a(cons_b(y0)))))))) ENCARG(cons_b(a(cons_b(a(cons_b(a(y0))))))) -> c1(ENCARG(a(cons_b(a(cons_b(a(y0))))))) ENCARG(cons_b(a(cons_b(a(cons_b(cons_b(a(y0)))))))) -> c1(ENCARG(a(cons_b(a(cons_b(cons_b(a(y0)))))))) ENCARG(cons_b(a(cons_b(a(cons_b(cons_b(cons_b(y0)))))))) -> c1(ENCARG(a(cons_b(a(cons_b(cons_b(cons_b(y0)))))))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) b(a(a(b(a(z0))))) -> a(a(b(b(a(a(b(z0))))))) Tuples: ENCARG(cons_b(a(a(z0)))) -> c1(B(a(a(encArg(z0)))), ENCARG(a(a(z0)))) ENCARG(cons_b(a(cons_b(z0)))) -> c1(B(a(b(encArg(z0)))), ENCARG(a(cons_b(z0)))) ENCARG(cons_b(cons_b(a(z0)))) -> c1(B(b(a(encArg(z0)))), ENCARG(cons_b(a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c1(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) ENCODE_B(a(a(z0))) -> c3(B(a(a(encArg(z0))))) ENCODE_B(a(cons_b(z0))) -> c3(B(a(b(encArg(z0))))) ENCODE_B(cons_b(a(z0))) -> c3(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c3(B(b(b(encArg(z0))))) ENCARG(a(cons_b(a(a(y0))))) -> c(ENCARG(cons_b(a(a(y0))))) ENCARG(a(cons_b(a(cons_b(y0))))) -> c(ENCARG(cons_b(a(cons_b(y0))))) ENCARG(a(cons_b(cons_b(a(y0))))) -> c(ENCARG(cons_b(cons_b(a(y0))))) ENCARG(a(cons_b(cons_b(cons_b(y0))))) -> c(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(cons_b(a(cons_b(a(a(y0)))))) -> c1(ENCARG(a(cons_b(a(a(y0)))))) ENCARG(cons_b(a(cons_b(a(cons_b(y0)))))) -> c1(ENCARG(a(cons_b(a(cons_b(y0)))))) ENCARG(cons_b(a(cons_b(cons_b(a(y0)))))) -> c1(ENCARG(a(cons_b(cons_b(a(y0)))))) ENCARG(cons_b(a(cons_b(cons_b(cons_b(y0)))))) -> c1(ENCARG(a(cons_b(cons_b(cons_b(y0)))))) B(a(a(b(a(a(a(b(a(a(a(y0))))))))))) -> c4(B(a(a(b(a(a(a(y0)))))))) B(a(a(b(a(a(a(b(a(a(a(a(b(a(y0)))))))))))))) -> c4(B(a(a(b(a(a(a(a(b(a(y0))))))))))) B(a(a(b(a(a(a(b(a(a(a(b(a(y0))))))))))))) -> c4(B(a(a(b(a(a(a(b(a(y0)))))))))) ENCARG(a(a(a(y0)))) -> c(ENCARG(a(a(y0)))) ENCARG(a(a(cons_b(a(a(y0)))))) -> c(ENCARG(a(cons_b(a(a(y0)))))) ENCARG(a(a(cons_b(a(cons_b(y0)))))) -> c(ENCARG(a(cons_b(a(cons_b(y0)))))) ENCARG(a(a(cons_b(a(y0))))) -> c(ENCARG(a(cons_b(a(y0))))) ENCARG(a(a(cons_b(cons_b(a(y0)))))) -> c(ENCARG(a(cons_b(cons_b(a(y0)))))) ENCARG(a(a(cons_b(cons_b(cons_b(y0)))))) -> c(ENCARG(a(cons_b(cons_b(cons_b(y0)))))) ENCARG(a(cons_b(a(cons_b(a(a(y0))))))) -> c(ENCARG(cons_b(a(cons_b(a(a(y0))))))) ENCARG(a(cons_b(a(cons_b(a(cons_b(y0))))))) -> c(ENCARG(cons_b(a(cons_b(a(cons_b(y0))))))) ENCARG(a(cons_b(a(cons_b(a(y0)))))) -> c(ENCARG(cons_b(a(cons_b(a(y0)))))) ENCARG(a(cons_b(a(cons_b(cons_b(a(y0))))))) -> c(ENCARG(cons_b(a(cons_b(cons_b(a(y0))))))) ENCARG(a(cons_b(a(cons_b(cons_b(cons_b(y0))))))) -> c(ENCARG(cons_b(a(cons_b(cons_b(cons_b(y0))))))) ENCARG(cons_b(a(a(a(y0))))) -> c1(ENCARG(a(a(a(y0))))) ENCARG(cons_b(a(a(cons_b(a(a(y0))))))) -> c1(ENCARG(a(a(cons_b(a(a(y0))))))) ENCARG(cons_b(a(a(cons_b(a(cons_b(y0))))))) -> c1(ENCARG(a(a(cons_b(a(cons_b(y0))))))) ENCARG(cons_b(a(a(cons_b(a(y0)))))) -> c1(ENCARG(a(a(cons_b(a(y0)))))) ENCARG(cons_b(a(a(cons_b(cons_b(a(y0))))))) -> c1(ENCARG(a(a(cons_b(cons_b(a(y0))))))) ENCARG(cons_b(a(a(cons_b(cons_b(cons_b(y0))))))) -> c1(ENCARG(a(a(cons_b(cons_b(cons_b(y0))))))) ENCARG(cons_b(a(cons_b(a(cons_b(a(a(y0)))))))) -> c1(ENCARG(a(cons_b(a(cons_b(a(a(y0)))))))) ENCARG(cons_b(a(cons_b(a(cons_b(a(cons_b(y0)))))))) -> c1(ENCARG(a(cons_b(a(cons_b(a(cons_b(y0)))))))) ENCARG(cons_b(a(cons_b(a(cons_b(a(y0))))))) -> c1(ENCARG(a(cons_b(a(cons_b(a(y0))))))) ENCARG(cons_b(a(cons_b(a(cons_b(cons_b(a(y0)))))))) -> c1(ENCARG(a(cons_b(a(cons_b(cons_b(a(y0)))))))) ENCARG(cons_b(a(cons_b(a(cons_b(cons_b(cons_b(y0)))))))) -> c1(ENCARG(a(cons_b(a(cons_b(cons_b(cons_b(y0)))))))) S tuples: B(a(a(b(a(a(a(z0))))))) -> c4(B(a(a(b(a(a(b(b(a(a(b(z0))))))))))), B(a(a(b(a(a(z0)))))), B(a(a(z0)))) B(a(a(b(a(a(a(a(b(a(z0)))))))))) -> c4(B(a(a(b(a(a(a(b(a(z0)))))))))) B(a(a(b(a(a(a(b(a(a(a(y0))))))))))) -> c4(B(a(a(b(a(a(a(y0)))))))) B(a(a(b(a(a(a(b(a(a(a(a(b(a(y0)))))))))))))) -> c4(B(a(a(b(a(a(a(a(b(a(y0))))))))))) B(a(a(b(a(a(a(b(a(a(a(b(a(y0))))))))))))) -> c4(B(a(a(b(a(a(a(b(a(y0)))))))))) K tuples:none Defined Rule Symbols: encArg_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c1_2, c4_3, c4_1, c3_1, c_1, c1_1