/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF).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), 469 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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) CompletionProof [UPPER BOUND(ID), 0 ms] (28) CpxTypedWeightedCompleteTrs (29) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CpxTypedWeightedCompleteTrs (31) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) SimplificationProof [BOTH BOUNDS(ID, 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), 6 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) 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) CdtNarrowingProof [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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) 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(b(b(x1))) -> a(a(a(x1))) a(a(a(x1))) -> b(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(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(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(b(b(x1))) -> a(a(a(x1))) a(a(a(x1))) -> b(a(b(x1))) The (relative) TRS S consists of the following rules: encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(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(b(b(x1))) -> a(a(a(x1))) a(a(a(x1))) -> b(a(b(x1))) The (relative) TRS S consists of the following rules: encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(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: b(b(b(x1))) -> a(a(a(x1))) a(a(a(x1))) -> b(a(b(x1))) The (relative) TRS S consists of the following rules: encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(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: b(b(b(x1))) -> a(a(a(x1))) a(a(a(x1))) -> b(a(b(x1))) The (relative) TRS S consists of the following rules: encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(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: b(b(b(x1))) -> a(a(a(x1))) a(a(a(x1))) -> b(a(b(x1))) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Types: b :: b:a:encArg:encode_b:encode_a -> b:a:encArg:encode_b:encode_a a :: b:a:encArg:encode_b:encode_a -> b:a:encArg:encode_b:encode_a encArg :: cons_b:cons_a -> b:a:encArg:encode_b:encode_a cons_b :: cons_b:cons_a -> cons_b:cons_a cons_a :: cons_b:cons_a -> cons_b:cons_a encode_b :: cons_b:cons_a -> b:a:encArg:encode_b:encode_a encode_a :: cons_b:cons_a -> b:a:encArg:encode_b:encode_a hole_b:a:encArg:encode_b:encode_a1_0 :: b:a:encArg:encode_b:encode_a hole_cons_b:cons_a2_0 :: cons_b:cons_a gen_cons_b:cons_a3_0 :: Nat -> cons_b:cons_a ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: b, a, encArg They will be analysed ascendingly in the following order: b = a b < encArg a < encArg ---------------------------------------- (12) Obligation: Innermost TRS: Rules: b(b(b(x1))) -> a(a(a(x1))) a(a(a(x1))) -> b(a(b(x1))) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Types: b :: b:a:encArg:encode_b:encode_a -> b:a:encArg:encode_b:encode_a a :: b:a:encArg:encode_b:encode_a -> b:a:encArg:encode_b:encode_a encArg :: cons_b:cons_a -> b:a:encArg:encode_b:encode_a cons_b :: cons_b:cons_a -> cons_b:cons_a cons_a :: cons_b:cons_a -> cons_b:cons_a encode_b :: cons_b:cons_a -> b:a:encArg:encode_b:encode_a encode_a :: cons_b:cons_a -> b:a:encArg:encode_b:encode_a hole_b:a:encArg:encode_b:encode_a1_0 :: b:a:encArg:encode_b:encode_a hole_cons_b:cons_a2_0 :: cons_b:cons_a gen_cons_b:cons_a3_0 :: Nat -> cons_b:cons_a Generator Equations: gen_cons_b:cons_a3_0(0) <=> hole_cons_b:cons_a2_0 gen_cons_b:cons_a3_0(+(x, 1)) <=> cons_b(gen_cons_b:cons_a3_0(x)) The following defined symbols remain to be analysed: a, b, encArg They will be analysed ascendingly in the following order: b = a b < encArg a < encArg ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_cons_b:cons_a3_0(+(1, n11_0))) -> *4_0, rt in Omega(0) Induction Base: encArg(gen_cons_b:cons_a3_0(+(1, 0))) Induction Step: encArg(gen_cons_b:cons_a3_0(+(1, +(n11_0, 1)))) ->_R^Omega(0) b(encArg(gen_cons_b:cons_a3_0(+(1, n11_0)))) ->_IH b(*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: b(b(b(x1))) -> a(a(a(x1))) a(a(a(x1))) -> b(a(b(x1))) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(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_a(c_a(x1))) -> b(a(b(x1))) b(c_b(c_b(x1))) -> a(a(a(x1))) The (relative) TRS S consists of the following rules: encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(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) Rewrite Strategy: INNERMOST ---------------------------------------- (19) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (20) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(c_a(c_a(x1))) -> b(a(b(x1))) [1] b(c_b(c_b(x1))) -> a(a(a(x1))) [1] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encArg(cons_a(x_1)) -> a(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] Rewrite Strategy: INNERMOST ---------------------------------------- (21) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (22) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(c_a(c_a(x1))) -> b(a(b(x1))) [1] b(c_b(c_b(x1))) -> a(a(a(x1))) [1] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encArg(cons_a(x_1)) -> a(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] The TRS has the following type information: a :: c_a:c_b -> c_a:c_b c_a :: c_a:c_b -> c_a:c_b b :: c_a:c_b -> c_a:c_b c_b :: c_a:c_b -> c_a:c_b encArg :: cons_b:cons_a -> c_a:c_b cons_b :: cons_b:cons_a -> cons_b:cons_a cons_a :: cons_b:cons_a -> cons_b:cons_a encode_b :: cons_b:cons_a -> c_a:c_b encode_a :: cons_b:cons_a -> c_a:c_b Rewrite Strategy: INNERMOST ---------------------------------------- (23) 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] a(v0) -> null_a [0] b(v0) -> null_b [0] And the following fresh constants: null_encArg, null_encode_b, null_encode_a, null_a, null_b, const ---------------------------------------- (24) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(c_a(c_a(x1))) -> b(a(b(x1))) [1] b(c_b(c_b(x1))) -> a(a(a(x1))) [1] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encArg(cons_a(x_1)) -> a(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] encArg(v0) -> null_encArg [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] The TRS has the following type information: a :: c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b -> c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b c_a :: c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b -> c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b b :: c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b -> c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b c_b :: c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b -> c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b encArg :: cons_b:cons_a -> c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b cons_b :: cons_b:cons_a -> cons_b:cons_a cons_a :: cons_b:cons_a -> cons_b:cons_a encode_b :: cons_b:cons_a -> c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b encode_a :: cons_b:cons_a -> c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b null_encArg :: c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b null_encode_b :: c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b null_encode_a :: c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b null_a :: c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b null_b :: c_a:c_b:null_encArg:null_encode_b:null_encode_a:null_a:null_b const :: cons_b:cons_a Rewrite Strategy: INNERMOST ---------------------------------------- (25) 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_a => 0 null_b => 0 const => 0 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 1 }-> b(a(b(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(x1))) :|: z = 1 + (1 + x1), x1 >= 0 b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 b(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 encArg(z) -{ 0 }-> b(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> a(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 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 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (27) 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 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] a(v0) -> const [0] b(v0) -> const [0] And the following fresh constants: const, const1 ---------------------------------------- (28) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(c_a(c_a(x1))) -> b(a(b(x1))) [1] b(c_b(c_b(x1))) -> a(a(a(x1))) [1] encArg(cons_b(x_1)) -> b(encArg(x_1)) [0] encArg(cons_a(x_1)) -> a(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] encArg(v0) -> const [0] encode_b(v0) -> const [0] encode_a(v0) -> const [0] a(v0) -> const [0] b(v0) -> const [0] The TRS has the following type information: a :: c_a:c_b:const -> c_a:c_b:const c_a :: c_a:c_b:const -> c_a:c_b:const b :: c_a:c_b:const -> c_a:c_b:const c_b :: c_a:c_b:const -> c_a:c_b:const encArg :: cons_b:cons_a -> c_a:c_b:const cons_b :: cons_b:cons_a -> cons_b:cons_a cons_a :: cons_b:cons_a -> cons_b:cons_a encode_b :: cons_b:cons_a -> c_a:c_b:const encode_a :: cons_b:cons_a -> c_a:c_b:const const :: c_a:c_b:const const1 :: cons_b:cons_a Rewrite Strategy: INNERMOST ---------------------------------------- (29) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (30) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(c_a(c_a(c_b(c_b(x1'))))) -> b(a(a(a(a(x1'))))) [2] a(c_a(c_a(x1))) -> b(a(c_b(x1))) [1] a(c_a(c_a(x1))) -> b(a(const)) [1] b(c_b(c_b(c_a(c_a(x1''))))) -> a(a(b(a(b(x1''))))) [2] b(c_b(c_b(x1))) -> a(a(c_a(x1))) [1] b(c_b(c_b(x1))) -> a(a(const)) [1] encArg(cons_b(cons_b(x_1'))) -> b(b(encArg(x_1'))) [0] encArg(cons_b(cons_a(x_1''))) -> b(a(encArg(x_1''))) [0] encArg(cons_b(x_1)) -> b(const) [0] encArg(cons_a(cons_b(x_11))) -> a(b(encArg(x_11))) [0] encArg(cons_a(cons_a(x_12))) -> a(a(encArg(x_12))) [0] encArg(cons_a(x_1)) -> a(const) [0] encode_b(cons_b(x_13)) -> b(b(encArg(x_13))) [0] encode_b(cons_a(x_14)) -> b(a(encArg(x_14))) [0] encode_b(x_1) -> b(const) [0] encode_a(cons_b(x_15)) -> a(b(encArg(x_15))) [0] encode_a(cons_a(x_16)) -> a(a(encArg(x_16))) [0] encode_a(x_1) -> a(const) [0] a(x0) -> c_a(x0) [0] b(x0) -> c_b(x0) [0] encArg(v0) -> const [0] encode_b(v0) -> const [0] encode_a(v0) -> const [0] a(v0) -> const [0] b(v0) -> const [0] The TRS has the following type information: a :: c_a:c_b:const -> c_a:c_b:const c_a :: c_a:c_b:const -> c_a:c_b:const b :: c_a:c_b:const -> c_a:c_b:const c_b :: c_a:c_b:const -> c_a:c_b:const encArg :: cons_b:cons_a -> c_a:c_b:const cons_b :: cons_b:cons_a -> cons_b:cons_a cons_a :: cons_b:cons_a -> cons_b:cons_a encode_b :: cons_b:cons_a -> c_a:c_b:const encode_a :: cons_b:cons_a -> c_a:c_b:const const :: c_a:c_b:const const1 :: cons_b:cons_a Rewrite Strategy: INNERMOST ---------------------------------------- (31) 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 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 2 }-> b(a(a(a(a(x1'))))) :|: x1' >= 0, z = 1 + (1 + (1 + (1 + x1'))) a(z) -{ 1 }-> b(a(0)) :|: z = 1 + (1 + x1), x1 >= 0 a(z) -{ 1 }-> b(a(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(b(a(b(x1''))))) :|: z = 1 + (1 + (1 + (1 + x1''))), x1'' >= 0 b(z) -{ 1 }-> a(a(0)) :|: z = 1 + (1 + x1), x1 >= 0 b(z) -{ 1 }-> a(a(1 + x1)) :|: z = 1 + (1 + x1), x1 >= 0 b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 b(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 encArg(z) -{ 0 }-> b(b(encArg(x_1'))) :|: z = 1 + (1 + x_1'), x_1' >= 0 encArg(z) -{ 0 }-> b(a(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 }-> a(b(encArg(x_11))) :|: x_11 >= 0, z = 1 + (1 + x_11) encArg(z) -{ 0 }-> a(a(encArg(x_12))) :|: z = 1 + (1 + x_12), x_12 >= 0 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(b(encArg(x_15))) :|: x_15 >= 0, z = 1 + x_15 encode_a(z) -{ 0 }-> a(a(encArg(x_16))) :|: z = 1 + x_16, x_16 >= 0 encode_a(z) -{ 0 }-> a(0) :|: x_1 >= 0, z = x_1 encode_a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_b(z) -{ 0 }-> b(b(encArg(x_13))) :|: z = 1 + x_13, x_13 >= 0 encode_b(z) -{ 0 }-> b(a(encArg(x_14))) :|: x_14 >= 0, z = 1 + x_14 encode_b(z) -{ 0 }-> b(0) :|: x_1 >= 0, z = x_1 encode_b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (33) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 2 }-> b(a(a(a(a(z - 4))))) :|: z - 4 >= 0 a(z) -{ 1 }-> b(a(0)) :|: z - 2 >= 0 a(z) -{ 1 }-> b(a(1 + (z - 2))) :|: z - 2 >= 0 a(z) -{ 0 }-> 0 :|: z >= 0 a(z) -{ 0 }-> 1 + z :|: z >= 0 b(z) -{ 2 }-> a(a(b(a(b(z - 4))))) :|: z - 4 >= 0 b(z) -{ 1 }-> a(a(0)) :|: z - 2 >= 0 b(z) -{ 1 }-> a(a(1 + (z - 2))) :|: z - 2 >= 0 b(z) -{ 0 }-> 0 :|: z >= 0 b(z) -{ 0 }-> 1 + z :|: z >= 0 encArg(z) -{ 0 }-> b(b(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> b(a(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> b(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> 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(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(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 ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: ENCARG(cons_b(z0)) -> c(B(encArg(z0)), ENCARG(z0)) ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCODE_B(z0) -> c2(B(encArg(z0)), ENCARG(z0)) ENCODE_A(z0) -> c3(A(encArg(z0)), ENCARG(z0)) B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(z0))) -> c5(B(a(b(z0))), A(b(z0)), B(z0)) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(z0))) -> c5(B(a(b(z0))), A(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: b_1, a_1, encArg_1, encode_b_1, encode_a_1 Defined Pair Symbols: ENCARG_1, ENCODE_B_1, ENCODE_A_1, B_1, A_1 Compound Symbols: c_2, c1_2, c2_2, c3_2, c4_3, c5_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_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: ENCARG(cons_b(z0)) -> c(B(encArg(z0)), ENCARG(z0)) ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(z0))) -> c5(B(a(b(z0))), A(b(z0)), B(z0)) ENCODE_B(z0) -> c6(B(encArg(z0))) ENCODE_B(z0) -> c6(ENCARG(z0)) ENCODE_A(z0) -> c6(A(encArg(z0))) ENCODE_A(z0) -> c6(ENCARG(z0)) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(z0))) -> c5(B(a(b(z0))), A(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: b_1, a_1, encArg_1, encode_b_1, encode_a_1 Defined Pair Symbols: ENCARG_1, B_1, A_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c_2, c1_2, c4_3, c5_3, c6_1 ---------------------------------------- (39) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ENCODE_B(z0) -> c6(ENCARG(z0)) ENCODE_A(z0) -> c6(ENCARG(z0)) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: ENCARG(cons_b(z0)) -> c(B(encArg(z0)), ENCARG(z0)) ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(z0))) -> c5(B(a(b(z0))), A(b(z0)), B(z0)) ENCODE_B(z0) -> c6(B(encArg(z0))) ENCODE_A(z0) -> c6(A(encArg(z0))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(z0))) -> c5(B(a(b(z0))), A(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: b_1, a_1, encArg_1, encode_b_1, encode_a_1 Defined Pair Symbols: ENCARG_1, B_1, A_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c_2, c1_2, c4_3, c5_3, c6_1 ---------------------------------------- (41) 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)) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: ENCARG(cons_b(z0)) -> c(B(encArg(z0)), ENCARG(z0)) ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(z0))) -> c5(B(a(b(z0))), A(b(z0)), B(z0)) ENCODE_B(z0) -> c6(B(encArg(z0))) ENCODE_A(z0) -> c6(A(encArg(z0))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(z0))) -> c5(B(a(b(z0))), A(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: ENCARG_1, B_1, A_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c_2, c1_2, c4_3, c5_3, c6_1 ---------------------------------------- (43) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(z0)) -> c(B(encArg(z0)), ENCARG(z0)) by ENCARG(cons_b(cons_b(z0))) -> c(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c(B(a(encArg(z0))), ENCARG(cons_a(z0))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(z0))) -> c5(B(a(b(z0))), A(b(z0)), B(z0)) ENCODE_B(z0) -> c6(B(encArg(z0))) ENCODE_A(z0) -> c6(A(encArg(z0))) ENCARG(cons_b(cons_b(z0))) -> c(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c(B(a(encArg(z0))), ENCARG(cons_a(z0))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(z0))) -> c5(B(a(b(z0))), A(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: ENCARG_1, B_1, A_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c1_2, c4_3, c5_3, c6_1, c_2 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) by ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(z0))) -> c5(B(a(b(z0))), A(b(z0)), B(z0)) ENCODE_B(z0) -> c6(B(encArg(z0))) ENCODE_A(z0) -> c6(A(encArg(z0))) ENCARG(cons_b(cons_b(z0))) -> c(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(z0))) -> c5(B(a(b(z0))), A(b(z0)), B(z0)) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: B_1, A_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1 Compound Symbols: c4_3, c5_3, c6_1, c_2, c1_2 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(a(a(z0))) -> c5(B(a(b(z0))), A(b(z0)), B(z0)) by A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) ENCODE_B(z0) -> c6(B(encArg(z0))) ENCODE_A(z0) -> c6(A(encArg(z0))) ENCARG(cons_b(cons_b(z0))) -> c(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: B_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, A_1 Compound Symbols: c4_3, c6_1, c_2, c1_2, c5_3 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_B(z0) -> c6(B(encArg(z0))) by ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) ENCODE_A(z0) -> c6(A(encArg(z0))) ENCARG(cons_b(cons_b(z0))) -> c(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: B_1, ENCODE_A_1, ENCARG_1, A_1, ENCODE_B_1 Compound Symbols: c4_3, c6_1, c_2, c1_2, c5_3 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_A(z0) -> c6(A(encArg(z0))) by ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) ENCARG(cons_b(cons_b(z0))) -> c(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(cons_a(z0))) -> c(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: B_1, ENCARG_1, A_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c4_3, c_2, c1_2, c5_3, c6_1 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(cons_b(z0))) -> c(B(b(encArg(z0))), ENCARG(cons_b(z0))) by ENCARG(cons_b(cons_b(cons_b(z0)))) -> c(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) ENCARG(cons_b(cons_a(z0))) -> c(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: B_1, ENCARG_1, A_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c4_3, c_2, c1_2, c5_3, c6_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(cons_a(z0))) -> c(B(a(encArg(z0))), ENCARG(cons_a(z0))) by ENCARG(cons_b(cons_a(cons_b(z0)))) -> c(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: B_1, ENCARG_1, A_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c4_3, c1_2, c5_3, c6_1, c_2 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_a(cons_b(z0))) -> c1(A(b(encArg(z0))), ENCARG(cons_b(z0))) by ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: B_1, ENCARG_1, A_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c4_3, c1_2, c5_3, c6_1, c_2 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_a(cons_a(z0))) -> c1(A(a(encArg(z0))), ENCARG(cons_a(z0))) by ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: B_1, A_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1 Compound Symbols: c4_3, c5_3, c6_1, c_2, c1_2 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(a(a(b(b(z0))))) -> c5(B(a(a(a(a(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) by A(a(a(b(b(x0))))) -> c5(B(b(a(b(a(x0))))), A(b(b(b(x0)))), B(b(b(x0)))) A(a(a(b(b(z0))))) -> c5(B(a(b(a(b(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) A(a(a(b(b(a(z0)))))) -> c5(B(a(a(b(a(b(z0)))))), A(b(b(b(a(z0))))), B(b(b(a(z0))))) A(a(a(b(b(a(a(z0))))))) -> c5(B(a(a(a(b(a(b(z0))))))), A(b(b(b(a(a(z0)))))), B(b(b(a(a(z0)))))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) A(a(a(b(b(x0))))) -> c5(B(b(a(b(a(x0))))), A(b(b(b(x0)))), B(b(b(x0)))) A(a(a(b(b(z0))))) -> c5(B(a(b(a(b(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) A(a(a(b(b(a(z0)))))) -> c5(B(a(a(b(a(b(z0)))))), A(b(b(b(a(z0))))), B(b(b(a(z0))))) A(a(a(b(b(a(a(z0))))))) -> c5(B(a(a(a(b(a(b(z0))))))), A(b(b(b(a(a(z0)))))), B(b(b(a(a(z0)))))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(b(b(x0))))) -> c5(B(b(a(b(a(x0))))), A(b(b(b(x0)))), B(b(b(x0)))) A(a(a(b(b(z0))))) -> c5(B(a(b(a(b(z0))))), A(b(b(b(z0)))), B(b(b(z0)))) A(a(a(b(b(a(z0)))))) -> c5(B(a(a(b(a(b(z0)))))), A(b(b(b(a(z0))))), B(b(b(a(z0))))) A(a(a(b(b(a(a(z0))))))) -> c5(B(a(a(a(b(a(b(z0))))))), A(b(b(b(a(a(z0)))))), B(b(b(a(a(z0)))))) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: B_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, A_1 Compound Symbols: c4_3, c6_1, c_2, c1_2, c5_3 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) A(a(a(b(b(x0))))) -> c5(B(b(a(b(a(x0))))), A(b(b(b(x0)))), B(b(b(x0)))) A(a(a(b(b(a(z0)))))) -> c5(B(a(a(b(a(b(z0)))))), A(b(b(b(a(z0))))), B(b(b(a(z0))))) A(a(a(b(b(a(a(z0))))))) -> c5(B(a(a(a(b(a(b(z0))))))), A(b(b(b(a(a(z0)))))), B(b(b(a(a(z0)))))) A(a(a(b(b(z0))))) -> c5(A(b(b(b(z0)))), B(b(b(z0)))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(b(b(x0))))) -> c5(B(b(a(b(a(x0))))), A(b(b(b(x0)))), B(b(b(x0)))) A(a(a(b(b(a(z0)))))) -> c5(B(a(a(b(a(b(z0)))))), A(b(b(b(a(z0))))), B(b(b(a(z0))))) A(a(a(b(b(a(a(z0))))))) -> c5(B(a(a(a(b(a(b(z0))))))), A(b(b(b(a(a(z0)))))), B(b(b(a(a(z0)))))) A(a(a(b(b(z0))))) -> c5(A(b(b(b(z0)))), B(b(b(z0)))) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: B_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, A_1 Compound Symbols: c4_3, c6_1, c_2, c1_2, c5_3, c5_2 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) by ENCODE_B(cons_b(cons_b(z0))) -> c6(B(b(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c6(B(b(a(encArg(z0))))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) A(a(a(b(b(x0))))) -> c5(B(b(a(b(a(x0))))), A(b(b(b(x0)))), B(b(b(x0)))) A(a(a(b(b(a(z0)))))) -> c5(B(a(a(b(a(b(z0)))))), A(b(b(b(a(z0))))), B(b(b(a(z0))))) A(a(a(b(b(a(a(z0))))))) -> c5(B(a(a(a(b(a(b(z0))))))), A(b(b(b(a(a(z0)))))), B(b(b(a(a(z0)))))) A(a(a(b(b(z0))))) -> c5(A(b(b(b(z0)))), B(b(b(z0)))) ENCODE_B(cons_b(cons_b(z0))) -> c6(B(b(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c6(B(b(a(encArg(z0))))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(b(b(x0))))) -> c5(B(b(a(b(a(x0))))), A(b(b(b(x0)))), B(b(b(x0)))) A(a(a(b(b(a(z0)))))) -> c5(B(a(a(b(a(b(z0)))))), A(b(b(b(a(z0))))), B(b(b(a(z0))))) A(a(a(b(b(a(a(z0))))))) -> c5(B(a(a(a(b(a(b(z0))))))), A(b(b(b(a(a(z0)))))), B(b(b(a(a(z0)))))) A(a(a(b(b(z0))))) -> c5(A(b(b(b(z0)))), B(b(b(z0)))) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: B_1, ENCODE_B_1, ENCODE_A_1, ENCARG_1, A_1 Compound Symbols: c4_3, c6_1, c_2, c1_2, c5_3, c5_2 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) by ENCODE_B(cons_a(cons_b(z0))) -> c6(B(a(b(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c6(B(a(a(encArg(z0))))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) A(a(a(b(b(x0))))) -> c5(B(b(a(b(a(x0))))), A(b(b(b(x0)))), B(b(b(x0)))) A(a(a(b(b(a(z0)))))) -> c5(B(a(a(b(a(b(z0)))))), A(b(b(b(a(z0))))), B(b(b(a(z0))))) A(a(a(b(b(a(a(z0))))))) -> c5(B(a(a(a(b(a(b(z0))))))), A(b(b(b(a(a(z0)))))), B(b(b(a(a(z0)))))) A(a(a(b(b(z0))))) -> c5(A(b(b(b(z0)))), B(b(b(z0)))) ENCODE_B(cons_b(cons_b(z0))) -> c6(B(b(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c6(B(b(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c6(B(a(b(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c6(B(a(a(encArg(z0))))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(b(b(x0))))) -> c5(B(b(a(b(a(x0))))), A(b(b(b(x0)))), B(b(b(x0)))) A(a(a(b(b(a(z0)))))) -> c5(B(a(a(b(a(b(z0)))))), A(b(b(b(a(z0))))), B(b(b(a(z0))))) A(a(a(b(b(a(a(z0))))))) -> c5(B(a(a(a(b(a(b(z0))))))), A(b(b(b(a(a(z0)))))), B(b(b(a(a(z0)))))) A(a(a(b(b(z0))))) -> c5(A(b(b(b(z0)))), B(b(b(z0)))) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: B_1, ENCODE_A_1, ENCARG_1, A_1, ENCODE_B_1 Compound Symbols: c4_3, c6_1, c_2, c1_2, c5_3, c5_2 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) by ENCODE_A(cons_b(cons_b(z0))) -> c6(A(b(b(encArg(z0))))) ENCODE_A(cons_b(cons_a(z0))) -> c6(A(b(a(encArg(z0))))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) A(a(a(b(b(x0))))) -> c5(B(b(a(b(a(x0))))), A(b(b(b(x0)))), B(b(b(x0)))) A(a(a(b(b(a(z0)))))) -> c5(B(a(a(b(a(b(z0)))))), A(b(b(b(a(z0))))), B(b(b(a(z0))))) A(a(a(b(b(a(a(z0))))))) -> c5(B(a(a(a(b(a(b(z0))))))), A(b(b(b(a(a(z0)))))), B(b(b(a(a(z0)))))) A(a(a(b(b(z0))))) -> c5(A(b(b(b(z0)))), B(b(b(z0)))) ENCODE_B(cons_b(cons_b(z0))) -> c6(B(b(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c6(B(b(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c6(B(a(b(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c6(B(a(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c6(A(b(b(encArg(z0))))) ENCODE_A(cons_b(cons_a(z0))) -> c6(A(b(a(encArg(z0))))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(b(b(x0))))) -> c5(B(b(a(b(a(x0))))), A(b(b(b(x0)))), B(b(b(x0)))) A(a(a(b(b(a(z0)))))) -> c5(B(a(a(b(a(b(z0)))))), A(b(b(b(a(z0))))), B(b(b(a(z0))))) A(a(a(b(b(a(a(z0))))))) -> c5(B(a(a(a(b(a(b(z0))))))), A(b(b(b(a(a(z0)))))), B(b(b(a(a(z0)))))) A(a(a(b(b(z0))))) -> c5(A(b(b(b(z0)))), B(b(b(z0)))) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: B_1, ENCODE_A_1, ENCARG_1, A_1, ENCODE_B_1 Compound Symbols: c4_3, c6_1, c_2, c1_2, c5_3, c5_2 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) by ENCODE_A(cons_a(cons_b(z0))) -> c6(A(a(b(encArg(z0))))) ENCODE_A(cons_a(cons_a(z0))) -> c6(A(a(a(encArg(z0))))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_b(z0)) -> b(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) b(b(b(z0))) -> a(a(a(z0))) a(a(a(z0))) -> b(a(b(z0))) Tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c1(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c1(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_a(cons_b(z0)))) -> c1(A(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_a(cons_a(cons_a(z0)))) -> c1(A(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) A(a(a(b(b(x0))))) -> c5(B(b(a(b(a(x0))))), A(b(b(b(x0)))), B(b(b(x0)))) A(a(a(b(b(a(z0)))))) -> c5(B(a(a(b(a(b(z0)))))), A(b(b(b(a(z0))))), B(b(b(a(z0))))) A(a(a(b(b(a(a(z0))))))) -> c5(B(a(a(a(b(a(b(z0))))))), A(b(b(b(a(a(z0)))))), B(b(b(a(a(z0)))))) A(a(a(b(b(z0))))) -> c5(A(b(b(b(z0)))), B(b(b(z0)))) ENCODE_B(cons_b(cons_b(z0))) -> c6(B(b(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c6(B(b(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c6(B(a(b(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c6(B(a(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c6(A(b(b(encArg(z0))))) ENCODE_A(cons_b(cons_a(z0))) -> c6(A(b(a(encArg(z0))))) ENCODE_A(cons_a(cons_b(z0))) -> c6(A(a(b(encArg(z0))))) ENCODE_A(cons_a(cons_a(z0))) -> c6(A(a(a(encArg(z0))))) S tuples: B(b(b(z0))) -> c4(A(a(a(z0))), A(a(z0)), A(z0)) A(a(a(b(b(x0))))) -> c5(B(b(a(b(a(x0))))), A(b(b(b(x0)))), B(b(b(x0)))) A(a(a(b(b(a(z0)))))) -> c5(B(a(a(b(a(b(z0)))))), A(b(b(b(a(z0))))), B(b(b(a(z0))))) A(a(a(b(b(a(a(z0))))))) -> c5(B(a(a(a(b(a(b(z0))))))), A(b(b(b(a(a(z0)))))), B(b(b(a(a(z0)))))) A(a(a(b(b(z0))))) -> c5(A(b(b(b(z0)))), B(b(b(z0)))) K tuples:none Defined Rule Symbols: encArg_1, b_1, a_1 Defined Pair Symbols: B_1, ENCARG_1, A_1, ENCODE_B_1, ENCODE_A_1 Compound Symbols: c4_3, c_2, c1_2, c5_3, c5_2, c6_1