/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF).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), 32 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), 6 ms] (10) typed CpxTrs (11) OrderProof [LOWER BOUND(ID), 0 ms] (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 423 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) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxWeightedTrs (23) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxTypedWeightedTrs (25) CompletionProof [UPPER BOUND(ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CpxTypedWeightedCompleteTrs (29) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 2 ms] (30) CpxRNTS (31) SimplificationProof [BOTH BOUNDS(ID, ID), 1 ms] (32) CpxRNTS (33) CompletionProof [UPPER BOUND(ID), 0 ms] (34) CpxTypedWeightedCompleteTrs (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (38) CdtProblem (39) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (40) CdtProblem (41) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (46) CdtProblem (47) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 2 ms] (86) CdtProblem (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (96) 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: 0(0(0(0(x1)))) -> 1(0(1(1(x1)))) 0(1(0(1(x1)))) -> 0(0(1(0(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(1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(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: 0(0(0(0(x1)))) -> 1(0(1(1(x1)))) 0(1(0(1(x1)))) -> 0(0(1(0(x1)))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(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: 0(0(0(0(x1)))) -> 1(0(1(1(x1)))) 0(1(0(1(x1)))) -> 0(0(1(0(x1)))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(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: 0(0(0(0(x1)))) -> 1(0(1(1(x1)))) 0(1(0(1(x1)))) -> 0(0(1(0(x1)))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(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: 0(0(0(0(x1)))) -> 1(0(1(1(x1)))) 0(1(0(1(x1)))) -> 0(0(1(0(x1)))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Innermost TRS: Rules: 0(0(0(0(x1)))) -> 1(0(1(1(x1)))) 0(1(0(1(x1)))) -> 0(0(1(0(x1)))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) Types: 0 :: 1:cons_0 -> 1:cons_0 1 :: 1:cons_0 -> 1:cons_0 encArg :: 1:cons_0 -> 1:cons_0 cons_0 :: 1:cons_0 -> 1:cons_0 encode_0 :: 1:cons_0 -> 1:cons_0 encode_1 :: 1:cons_0 -> 1:cons_0 hole_1:cons_01_2 :: 1:cons_0 gen_1:cons_02_2 :: Nat -> 1:cons_0 ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: 0, encArg They will be analysed ascendingly in the following order: 0 < encArg ---------------------------------------- (12) Obligation: Innermost TRS: Rules: 0(0(0(0(x1)))) -> 1(0(1(1(x1)))) 0(1(0(1(x1)))) -> 0(0(1(0(x1)))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) Types: 0 :: 1:cons_0 -> 1:cons_0 1 :: 1:cons_0 -> 1:cons_0 encArg :: 1:cons_0 -> 1:cons_0 cons_0 :: 1:cons_0 -> 1:cons_0 encode_0 :: 1:cons_0 -> 1:cons_0 encode_1 :: 1:cons_0 -> 1:cons_0 hole_1:cons_01_2 :: 1:cons_0 gen_1:cons_02_2 :: Nat -> 1:cons_0 Generator Equations: gen_1:cons_02_2(0) <=> hole_1:cons_01_2 gen_1:cons_02_2(+(x, 1)) <=> 1(gen_1:cons_02_2(x)) The following defined symbols remain to be analysed: 0, encArg They will be analysed ascendingly in the following order: 0 < encArg ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_1:cons_02_2(+(1, n12_2))) -> *3_2, rt in Omega(0) Induction Base: encArg(gen_1:cons_02_2(+(1, 0))) Induction Step: encArg(gen_1:cons_02_2(+(1, +(n12_2, 1)))) ->_R^Omega(0) 1(encArg(gen_1:cons_02_2(+(1, n12_2)))) ->_IH 1(*3_2) 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: 0(0(0(0(x1)))) -> 1(0(1(1(x1)))) 0(1(0(1(x1)))) -> 0(0(1(0(x1)))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(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: 0(1(c_0(1(x1)))) -> 0(0(1(0(x1)))) 0(c_0(c_0(c_0(x1)))) -> 1(0(1(1(x1)))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) 0(x0) -> c_0(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: 0(1(c_0(1(x1)))) -> 0(0(1(0(x1)))) [1] 0(c_0(c_0(c_0(x1)))) -> 1(0(1(1(x1)))) [1] encArg(1(x_1)) -> 1(encArg(x_1)) [0] encArg(cons_0(x_1)) -> 0(encArg(x_1)) [0] encode_0(x_1) -> 0(encArg(x_1)) [0] encode_1(x_1) -> 1(encArg(x_1)) [0] 0(x0) -> c_0(x0) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (21) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: 0 => 0' ---------------------------------------- (22) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: 0'(1(c_0(1(x1)))) -> 0'(0'(1(0'(x1)))) [1] 0'(c_0(c_0(c_0(x1)))) -> 1(0'(1(1(x1)))) [1] encArg(1(x_1)) -> 1(encArg(x_1)) [0] encArg(cons_0(x_1)) -> 0'(encArg(x_1)) [0] encode_0(x_1) -> 0'(encArg(x_1)) [0] encode_1(x_1) -> 1(encArg(x_1)) [0] 0'(x0) -> c_0(x0) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (23) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (24) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 0'(1(c_0(1(x1)))) -> 0'(0'(1(0'(x1)))) [1] 0'(c_0(c_0(c_0(x1)))) -> 1(0'(1(1(x1)))) [1] encArg(1(x_1)) -> 1(encArg(x_1)) [0] encArg(cons_0(x_1)) -> 0'(encArg(x_1)) [0] encode_0(x_1) -> 0'(encArg(x_1)) [0] encode_1(x_1) -> 1(encArg(x_1)) [0] 0'(x0) -> c_0(x0) [0] The TRS has the following type information: 0' :: 1:c_0:cons_0 -> 1:c_0:cons_0 1 :: 1:c_0:cons_0 -> 1:c_0:cons_0 c_0 :: 1:c_0:cons_0 -> 1:c_0:cons_0 encArg :: 1:c_0:cons_0 -> 1:c_0:cons_0 cons_0 :: 1:c_0:cons_0 -> 1:c_0:cons_0 encode_0 :: 1:c_0:cons_0 -> 1:c_0:cons_0 encode_1 :: 1:c_0:cons_0 -> 1:c_0:cons_0 Rewrite Strategy: INNERMOST ---------------------------------------- (25) 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_0_1 encode_1_1 0'_1 Due to the following rules being added: encArg(v0) -> const [0] encode_0(v0) -> const [0] encode_1(v0) -> const [0] 0'(v0) -> const [0] And the following fresh constants: const ---------------------------------------- (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: 0'(1(c_0(1(x1)))) -> 0'(0'(1(0'(x1)))) [1] 0'(c_0(c_0(c_0(x1)))) -> 1(0'(1(1(x1)))) [1] encArg(1(x_1)) -> 1(encArg(x_1)) [0] encArg(cons_0(x_1)) -> 0'(encArg(x_1)) [0] encode_0(x_1) -> 0'(encArg(x_1)) [0] encode_1(x_1) -> 1(encArg(x_1)) [0] 0'(x0) -> c_0(x0) [0] encArg(v0) -> const [0] encode_0(v0) -> const [0] encode_1(v0) -> const [0] 0'(v0) -> const [0] The TRS has the following type information: 0' :: 1:c_0:cons_0:const -> 1:c_0:cons_0:const 1 :: 1:c_0:cons_0:const -> 1:c_0:cons_0:const c_0 :: 1:c_0:cons_0:const -> 1:c_0:cons_0:const encArg :: 1:c_0:cons_0:const -> 1:c_0:cons_0:const cons_0 :: 1:c_0:cons_0:const -> 1:c_0:cons_0:const encode_0 :: 1:c_0:cons_0:const -> 1:c_0:cons_0:const encode_1 :: 1:c_0:cons_0:const -> 1:c_0:cons_0:const const :: 1:c_0:cons_0:const Rewrite Strategy: INNERMOST ---------------------------------------- (27) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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: 0'(1(c_0(1(1(c_0(1(x1'))))))) -> 0'(0'(1(0'(0'(1(0'(x1'))))))) [2] 0'(1(c_0(1(c_0(c_0(c_0(x1''))))))) -> 0'(0'(1(1(0'(1(1(x1''))))))) [2] 0'(1(c_0(1(x1)))) -> 0'(0'(1(c_0(x1)))) [1] 0'(1(c_0(1(x1)))) -> 0'(0'(1(const))) [1] 0'(c_0(c_0(c_0(x1)))) -> 1(0'(1(1(x1)))) [1] encArg(1(x_1)) -> 1(encArg(x_1)) [0] encArg(cons_0(1(x_1'))) -> 0'(1(encArg(x_1'))) [0] encArg(cons_0(cons_0(x_1''))) -> 0'(0'(encArg(x_1''))) [0] encArg(cons_0(x_1)) -> 0'(const) [0] encode_0(1(x_11)) -> 0'(1(encArg(x_11))) [0] encode_0(cons_0(x_12)) -> 0'(0'(encArg(x_12))) [0] encode_0(x_1) -> 0'(const) [0] encode_1(x_1) -> 1(encArg(x_1)) [0] 0'(x0) -> c_0(x0) [0] encArg(v0) -> const [0] encode_0(v0) -> const [0] encode_1(v0) -> const [0] 0'(v0) -> const [0] The TRS has the following type information: 0' :: 1:c_0:cons_0:const -> 1:c_0:cons_0:const 1 :: 1:c_0:cons_0:const -> 1:c_0:cons_0:const c_0 :: 1:c_0:cons_0:const -> 1:c_0:cons_0:const encArg :: 1:c_0:cons_0:const -> 1:c_0:cons_0:const cons_0 :: 1:c_0:cons_0:const -> 1:c_0:cons_0:const encode_0 :: 1:c_0:cons_0:const -> 1:c_0:cons_0:const encode_1 :: 1:c_0:cons_0:const -> 1:c_0:cons_0:const const :: 1:c_0:cons_0:const Rewrite Strategy: INNERMOST ---------------------------------------- (29) 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 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: 0'(z) -{ 2 }-> 0'(0'(1 + 0'(0'(1 + 0'(x1'))))) :|: z = 1 + (1 + (1 + (1 + (1 + (1 + x1'))))), x1' >= 0 0'(z) -{ 1 }-> 0'(0'(1 + 0)) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) 0'(z) -{ 1 }-> 0'(0'(1 + (1 + x1))) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) 0'(z) -{ 2 }-> 0'(0'(1 + (1 + 0'(1 + (1 + x1''))))) :|: x1'' >= 0, z = 1 + (1 + (1 + (1 + (1 + (1 + x1''))))) 0'(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 0'(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 0'(z) -{ 1 }-> 1 + 0'(1 + (1 + x1)) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) encArg(z) -{ 0 }-> 0'(0'(encArg(x_1''))) :|: z = 1 + (1 + x_1''), x_1'' >= 0 encArg(z) -{ 0 }-> 0'(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> 0'(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_0(z) -{ 0 }-> 0'(0'(encArg(x_12))) :|: z = 1 + x_12, x_12 >= 0 encode_0(z) -{ 0 }-> 0'(0) :|: x_1 >= 0, z = x_1 encode_0(z) -{ 0 }-> 0'(1 + encArg(x_11)) :|: x_11 >= 0, z = 1 + x_11 encode_0(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_1(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1 ---------------------------------------- (31) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: 0'(z) -{ 2 }-> 0'(0'(1 + 0'(0'(1 + 0'(z - 6))))) :|: z - 6 >= 0 0'(z) -{ 1 }-> 0'(0'(1 + 0)) :|: z - 3 >= 0 0'(z) -{ 2 }-> 0'(0'(1 + (1 + 0'(1 + (1 + (z - 6)))))) :|: z - 6 >= 0 0'(z) -{ 1 }-> 0'(0'(1 + (1 + (z - 3)))) :|: z - 3 >= 0 0'(z) -{ 0 }-> 0 :|: z >= 0 0'(z) -{ 0 }-> 1 + z :|: z >= 0 0'(z) -{ 1 }-> 1 + 0'(1 + (1 + (z - 3))) :|: z - 3 >= 0 encArg(z) -{ 0 }-> 0'(0'(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> 0'(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> 0'(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0(z) -{ 0 }-> 0'(0'(encArg(z - 1))) :|: z - 1 >= 0 encode_0(z) -{ 0 }-> 0'(0) :|: z >= 0 encode_0(z) -{ 0 }-> 0'(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_0(z) -{ 0 }-> 0 :|: z >= 0 encode_1(z) -{ 0 }-> 0 :|: z >= 0 encode_1(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 ---------------------------------------- (33) 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_0(v0) -> null_encode_0 [0] encode_1(v0) -> null_encode_1 [0] 0'(v0) -> null_0' [0] And the following fresh constants: null_encArg, null_encode_0, null_encode_1, null_0' ---------------------------------------- (34) 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: 0'(1(c_0(1(x1)))) -> 0'(0'(1(0'(x1)))) [1] 0'(c_0(c_0(c_0(x1)))) -> 1(0'(1(1(x1)))) [1] encArg(1(x_1)) -> 1(encArg(x_1)) [0] encArg(cons_0(x_1)) -> 0'(encArg(x_1)) [0] encode_0(x_1) -> 0'(encArg(x_1)) [0] encode_1(x_1) -> 1(encArg(x_1)) [0] 0'(x0) -> c_0(x0) [0] encArg(v0) -> null_encArg [0] encode_0(v0) -> null_encode_0 [0] encode_1(v0) -> null_encode_1 [0] 0'(v0) -> null_0' [0] The TRS has the following type information: 0' :: 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' -> 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' 1 :: 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' -> 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' c_0 :: 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' -> 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' encArg :: 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' -> 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' cons_0 :: 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' -> 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' encode_0 :: 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' -> 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' encode_1 :: 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' -> 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' null_encArg :: 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' null_encode_0 :: 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' null_encode_1 :: 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' null_0' :: 1:c_0:cons_0:null_encArg:null_encode_0:null_encode_1:null_0' Rewrite Strategy: INNERMOST ---------------------------------------- (35) 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_0 => 0 null_encode_1 => 0 null_0' => 0 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: 0'(z) -{ 1 }-> 0'(0'(1 + 0'(x1))) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) 0'(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 0'(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 0'(z) -{ 1 }-> 1 + 0'(1 + (1 + x1)) :|: x1 >= 0, z = 1 + (1 + (1 + x1)) encArg(z) -{ 0 }-> 0'(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_0(z) -{ 0 }-> 0'(encArg(x_1)) :|: x_1 >= 0, z = x_1 encode_0(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_1(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (37) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) encode_0(z0) -> 0(encArg(z0)) encode_1(z0) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) ENCARG(cons_0(z0)) -> c1(0'(encArg(z0)), ENCARG(z0)) ENCODE_0(z0) -> c2(0'(encArg(z0)), ENCARG(z0)) ENCODE_1(z0) -> c3(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4(0'(1(1(z0)))) 0'(1(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 0'(z0)) S tuples: 0'(0(0(0(z0)))) -> c4(0'(1(1(z0)))) 0'(1(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 0'(z0)) K tuples:none Defined Rule Symbols: 0_1, encArg_1, encode_0_1, encode_1_1 Defined Pair Symbols: ENCARG_1, ENCODE_0_1, ENCODE_1_1, 0'_1 Compound Symbols: c_1, c1_2, c2_2, c3_1, c4_1, c5_3 ---------------------------------------- (39) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ENCODE_1(z0) -> c3(ENCARG(z0)) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) encode_0(z0) -> 0(encArg(z0)) encode_1(z0) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) ENCARG(cons_0(z0)) -> c1(0'(encArg(z0)), ENCARG(z0)) ENCODE_0(z0) -> c2(0'(encArg(z0)), ENCARG(z0)) 0'(0(0(0(z0)))) -> c4(0'(1(1(z0)))) 0'(1(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 0'(z0)) S tuples: 0'(0(0(0(z0)))) -> c4(0'(1(1(z0)))) 0'(1(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 0'(z0)) K tuples:none Defined Rule Symbols: 0_1, encArg_1, encode_0_1, encode_1_1 Defined Pair Symbols: ENCARG_1, ENCODE_0_1, 0'_1 Compound Symbols: c_1, c1_2, c2_2, c4_1, c5_3 ---------------------------------------- (41) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) encode_0(z0) -> 0(encArg(z0)) encode_1(z0) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) ENCARG(cons_0(z0)) -> c1(0'(encArg(z0)), ENCARG(z0)) ENCODE_0(z0) -> c2(0'(encArg(z0)), ENCARG(z0)) 0'(1(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 0'(z0)) 0'(0(0(0(z0)))) -> c4 S tuples: 0'(1(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 0'(z0)) 0'(0(0(0(z0)))) -> c4 K tuples:none Defined Rule Symbols: 0_1, encArg_1, encode_0_1, encode_1_1 Defined Pair Symbols: ENCARG_1, ENCODE_0_1, 0'_1 Compound Symbols: c_1, c1_2, c2_2, c5_3, c4 ---------------------------------------- (43) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) encode_0(z0) -> 0(encArg(z0)) encode_1(z0) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) ENCARG(cons_0(z0)) -> c1(0'(encArg(z0)), ENCARG(z0)) 0'(1(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 0'(z0)) 0'(0(0(0(z0)))) -> c4 ENCODE_0(z0) -> c3(0'(encArg(z0))) ENCODE_0(z0) -> c3(ENCARG(z0)) S tuples: 0'(1(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 0'(z0)) 0'(0(0(0(z0)))) -> c4 K tuples:none Defined Rule Symbols: 0_1, encArg_1, encode_0_1, encode_1_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c1_2, c5_3, c4, c3_1 ---------------------------------------- (45) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ENCODE_0(z0) -> c3(ENCARG(z0)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) encode_0(z0) -> 0(encArg(z0)) encode_1(z0) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) ENCARG(cons_0(z0)) -> c1(0'(encArg(z0)), ENCARG(z0)) 0'(1(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 0'(z0)) 0'(0(0(0(z0)))) -> c4 ENCODE_0(z0) -> c3(0'(encArg(z0))) S tuples: 0'(1(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 0'(z0)) 0'(0(0(0(z0)))) -> c4 K tuples:none Defined Rule Symbols: 0_1, encArg_1, encode_0_1, encode_1_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c1_2, c5_3, c4, c3_1 ---------------------------------------- (47) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_0(z0) -> 0(encArg(z0)) encode_1(z0) -> 1(encArg(z0)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) ENCARG(cons_0(z0)) -> c1(0'(encArg(z0)), ENCARG(z0)) 0'(1(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 0'(z0)) 0'(0(0(0(z0)))) -> c4 ENCODE_0(z0) -> c3(0'(encArg(z0))) S tuples: 0'(1(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 0'(z0)) 0'(0(0(0(z0)))) -> c4 K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c1_2, c5_3, c4, c3_1 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_0(z0)) -> c1(0'(encArg(z0)), ENCARG(z0)) by ENCARG(cons_0(1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(1(z0))) ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(1(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 0'(z0)) 0'(0(0(0(z0)))) -> c4 ENCODE_0(z0) -> c3(0'(encArg(z0))) ENCARG(cons_0(1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(1(z0))) ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0))) S tuples: 0'(1(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 0'(z0)) 0'(0(0(0(z0)))) -> c4 K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c5_3, c4, c3_1, c1_2 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace 0'(1(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 0'(z0)) by 0'(1(0(1(1(z0))))) -> c5(0'(0(0(1(0(z0))))), 0'(1(0(1(z0)))), 0'(1(z0))) 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(1(1(0(1(1(z0))))))), 0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4 ENCODE_0(z0) -> c3(0'(encArg(z0))) ENCARG(cons_0(1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(1(z0))) ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0))) 0'(1(0(1(1(z0))))) -> c5(0'(0(0(1(0(z0))))), 0'(1(0(1(z0)))), 0'(1(z0))) 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(1(1(0(1(1(z0))))))), 0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(z0))))) -> c5(0'(0(0(1(0(z0))))), 0'(1(0(1(z0)))), 0'(1(z0))) 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(1(1(0(1(1(z0))))))), 0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c4, c3_1, c1_2, c5_3 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4 ENCODE_0(z0) -> c3(0'(encArg(z0))) ENCARG(cons_0(1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(1(z0))) ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0))) 0'(1(0(1(1(z0))))) -> c5(0'(0(0(1(0(z0))))), 0'(1(0(1(z0)))), 0'(1(z0))) 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(z0))))) -> c5(0'(0(0(1(0(z0))))), 0'(1(0(1(z0)))), 0'(1(z0))) 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c4, c3_1, c1_2, c5_3, c5_2 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_0(z0) -> c3(0'(encArg(z0))) by ENCODE_0(1(z0)) -> c3(0'(1(encArg(z0)))) ENCODE_0(cons_0(z0)) -> c3(0'(0(encArg(z0)))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4 ENCARG(cons_0(1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(1(z0))) ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0))) 0'(1(0(1(1(z0))))) -> c5(0'(0(0(1(0(z0))))), 0'(1(0(1(z0)))), 0'(1(z0))) 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) ENCODE_0(1(z0)) -> c3(0'(1(encArg(z0)))) ENCODE_0(cons_0(z0)) -> c3(0'(0(encArg(z0)))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(z0))))) -> c5(0'(0(0(1(0(z0))))), 0'(1(0(1(z0)))), 0'(1(z0))) 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c4, c1_2, c5_3, c5_2, c3_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_0(1(z0))) -> c1(0'(1(encArg(z0))), ENCARG(1(z0))) by ENCARG(cons_0(1(1(z0)))) -> c1(0'(1(1(encArg(z0)))), ENCARG(1(1(z0)))) ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(x0))) -> c1(ENCARG(1(x0))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4 ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0))) 0'(1(0(1(1(z0))))) -> c5(0'(0(0(1(0(z0))))), 0'(1(0(1(z0)))), 0'(1(z0))) 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) ENCODE_0(1(z0)) -> c3(0'(1(encArg(z0)))) ENCODE_0(cons_0(z0)) -> c3(0'(0(encArg(z0)))) ENCARG(cons_0(1(1(z0)))) -> c1(0'(1(1(encArg(z0)))), ENCARG(1(1(z0)))) ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(x0))) -> c1(ENCARG(1(x0))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(z0))))) -> c5(0'(0(0(1(0(z0))))), 0'(1(0(1(z0)))), 0'(1(z0))) 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c4, c1_2, c5_3, c5_2, c3_1, c1_1 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4 ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0))) 0'(1(0(1(1(z0))))) -> c5(0'(0(0(1(0(z0))))), 0'(1(0(1(z0)))), 0'(1(z0))) 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) ENCODE_0(1(z0)) -> c3(0'(1(encArg(z0)))) ENCODE_0(cons_0(z0)) -> c3(0'(0(encArg(z0)))) ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(x0))) -> c1(ENCARG(1(x0))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(z0))))) -> c5(0'(0(0(1(0(z0))))), 0'(1(0(1(z0)))), 0'(1(z0))) 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c4, c1_2, c5_3, c5_2, c3_1, c1_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_0(cons_0(z0))) -> c1(0'(0(encArg(z0))), ENCARG(cons_0(z0))) by ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(z0))))) -> c5(0'(0(0(1(0(z0))))), 0'(1(0(1(z0)))), 0'(1(z0))) 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) ENCODE_0(1(z0)) -> c3(0'(1(encArg(z0)))) ENCODE_0(cons_0(z0)) -> c3(0'(0(encArg(z0)))) ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(x0))) -> c1(ENCARG(1(x0))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(z0))))) -> c5(0'(0(0(1(0(z0))))), 0'(1(0(1(z0)))), 0'(1(z0))) 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c4, c5_3, c5_2, c3_1, c1_2, c1_1 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace 0'(1(0(1(1(z0))))) -> c5(0'(0(0(1(0(z0))))), 0'(1(0(1(z0)))), 0'(1(z0))) by 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0))))), 0'(1(1(z0)))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(0(0(1(1(0(1(1(z0)))))))), 0'(1(0(1(0(0(0(z0))))))), 0'(1(0(0(0(z0)))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0))))))), 0'(1(1(0(1(z0)))))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) ENCODE_0(1(z0)) -> c3(0'(1(encArg(z0)))) ENCODE_0(cons_0(z0)) -> c3(0'(0(encArg(z0)))) ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(x0))) -> c1(ENCARG(1(x0))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0))))), 0'(1(1(z0)))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(0(0(1(1(0(1(1(z0)))))))), 0'(1(0(1(0(0(0(z0))))))), 0'(1(0(0(0(z0)))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0))))))), 0'(1(1(0(1(z0)))))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0))))), 0'(1(1(z0)))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(0(0(1(1(0(1(1(z0)))))))), 0'(1(0(1(0(0(0(z0))))))), 0'(1(0(0(0(z0)))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0))))))), 0'(1(1(0(1(z0)))))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c4, c5_3, c5_2, c3_1, c1_2, c1_1 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) ENCODE_0(1(z0)) -> c3(0'(1(encArg(z0)))) ENCODE_0(cons_0(z0)) -> c3(0'(0(encArg(z0)))) ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(x0))) -> c1(ENCARG(1(x0))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c4, c5_3, c5_2, c3_1, c1_2, c1_1, c5_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace 0'(1(0(1(1(0(1(z0))))))) -> c5(0'(0(1(0(0(1(0(z0))))))), 0'(1(0(1(0(1(z0)))))), 0'(1(0(1(z0))))) by 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(0(1(0(0(1(1(0(1(1(z0)))))))))), 0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) ENCODE_0(1(z0)) -> c3(0'(1(encArg(z0)))) ENCODE_0(cons_0(z0)) -> c3(0'(0(encArg(z0)))) ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(x0))) -> c1(ENCARG(1(x0))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(0(1(0(0(1(1(0(1(1(z0)))))))))), 0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(0(1(0(0(1(1(0(1(1(z0)))))))))), 0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c4, c5_2, c3_1, c1_2, c1_1, c5_1, c5_3 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) ENCODE_0(1(z0)) -> c3(0'(1(encArg(z0)))) ENCODE_0(cons_0(z0)) -> c3(0'(0(encArg(z0)))) ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(x0))) -> c1(ENCARG(1(x0))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c4, c5_2, c3_1, c1_2, c1_1, c5_1, c5_3 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 0'(0(0(0(z0))))) by 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(1(0(1(1(z0)))))), 0'(0(0(0(z0))))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4 ENCODE_0(1(z0)) -> c3(0'(1(encArg(z0)))) ENCODE_0(cons_0(z0)) -> c3(0'(0(encArg(z0)))) ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(x0))) -> c1(ENCARG(1(x0))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(1(0(1(1(z0)))))), 0'(0(0(0(z0))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(1(1(0(1(1(z0)))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c4, c3_1, c1_2, c1_1, c5_2, c5_1, c5_3 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4 ENCODE_0(1(z0)) -> c3(0'(1(encArg(z0)))) ENCODE_0(cons_0(z0)) -> c3(0'(0(encArg(z0)))) ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(x0))) -> c1(ENCARG(1(x0))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c4, c3_1, c1_2, c1_1, c5_2, c5_1, c5_3 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_0(1(z0)) -> c3(0'(1(encArg(z0)))) by ENCODE_0(1(1(z0))) -> c3(0'(1(1(encArg(z0))))) ENCODE_0(1(cons_0(z0))) -> c3(0'(1(0(encArg(z0))))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4 ENCODE_0(cons_0(z0)) -> c3(0'(0(encArg(z0)))) ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(x0))) -> c1(ENCARG(1(x0))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) ENCODE_0(1(1(z0))) -> c3(0'(1(1(encArg(z0))))) ENCODE_0(1(cons_0(z0))) -> c3(0'(1(0(encArg(z0))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c4, c3_1, c1_2, c1_1, c5_2, c5_1, c5_3 ---------------------------------------- (77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_0(1(1(z0))) -> c3(0'(1(1(encArg(z0))))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4 ENCODE_0(cons_0(z0)) -> c3(0'(0(encArg(z0)))) ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(x0))) -> c1(ENCARG(1(x0))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) ENCODE_0(1(cons_0(z0))) -> c3(0'(1(0(encArg(z0))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c4, c3_1, c1_2, c1_1, c5_2, c5_1, c5_3 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_0(cons_0(z0)) -> c3(0'(0(encArg(z0)))) by ENCODE_0(cons_0(1(z0))) -> c3(0'(0(1(encArg(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c3(0'(0(0(encArg(z0))))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(1(z0)) -> c(ENCARG(z0)) 0'(0(0(0(z0)))) -> c4 ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(x0))) -> c1(ENCARG(1(x0))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) ENCODE_0(1(cons_0(z0))) -> c3(0'(1(0(encArg(z0))))) ENCODE_0(cons_0(1(z0))) -> c3(0'(0(1(encArg(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c3(0'(0(0(encArg(z0))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: ENCARG_1, 0'_1, ENCODE_0_1 Compound Symbols: c_1, c4, c1_2, c1_1, c5_2, c5_1, c5_3, c3_1 ---------------------------------------- (81) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(1(z0)) -> c(ENCARG(z0)) by ENCARG(1(1(y0))) -> c(ENCARG(1(y0))) ENCARG(1(cons_0(1(cons_0(y0))))) -> c(ENCARG(cons_0(1(cons_0(y0))))) ENCARG(1(cons_0(1(y0)))) -> c(ENCARG(cons_0(1(y0)))) ENCARG(1(cons_0(1(1(y0))))) -> c(ENCARG(cons_0(1(1(y0))))) ENCARG(1(cons_0(cons_0(1(y0))))) -> c(ENCARG(cons_0(cons_0(1(y0))))) ENCARG(1(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4 ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(x0))) -> c1(ENCARG(1(x0))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) ENCODE_0(1(cons_0(z0))) -> c3(0'(1(0(encArg(z0))))) ENCODE_0(cons_0(1(z0))) -> c3(0'(0(1(encArg(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c3(0'(0(0(encArg(z0))))) ENCARG(1(1(y0))) -> c(ENCARG(1(y0))) ENCARG(1(cons_0(1(cons_0(y0))))) -> c(ENCARG(cons_0(1(cons_0(y0))))) ENCARG(1(cons_0(1(y0)))) -> c(ENCARG(cons_0(1(y0)))) ENCARG(1(cons_0(1(1(y0))))) -> c(ENCARG(cons_0(1(1(y0))))) ENCARG(1(cons_0(cons_0(1(y0))))) -> c(ENCARG(cons_0(cons_0(1(y0))))) ENCARG(1(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: 0'_1, ENCARG_1, ENCODE_0_1 Compound Symbols: c4, c1_2, c1_1, c5_2, c5_1, c5_3, c3_1, c_1 ---------------------------------------- (83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_0(1(x0))) -> c1(ENCARG(1(x0))) by ENCARG(cons_0(1(1(y0)))) -> c1(ENCARG(1(1(y0)))) ENCARG(cons_0(1(cons_0(1(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(cons_0(1(cons_0(1(y0))))) -> c1(ENCARG(1(cons_0(1(y0))))) ENCARG(cons_0(1(cons_0(1(1(y0)))))) -> c1(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(1(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(cons_0(y0)))))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4 ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) ENCODE_0(1(cons_0(z0))) -> c3(0'(1(0(encArg(z0))))) ENCODE_0(cons_0(1(z0))) -> c3(0'(0(1(encArg(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c3(0'(0(0(encArg(z0))))) ENCARG(1(1(y0))) -> c(ENCARG(1(y0))) ENCARG(1(cons_0(1(cons_0(y0))))) -> c(ENCARG(cons_0(1(cons_0(y0))))) ENCARG(1(cons_0(1(y0)))) -> c(ENCARG(cons_0(1(y0)))) ENCARG(1(cons_0(1(1(y0))))) -> c(ENCARG(cons_0(1(1(y0))))) ENCARG(1(cons_0(cons_0(1(y0))))) -> c(ENCARG(cons_0(cons_0(1(y0))))) ENCARG(1(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) ENCARG(cons_0(1(cons_0(1(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(cons_0(1(cons_0(1(y0))))) -> c1(ENCARG(1(cons_0(1(y0))))) ENCARG(cons_0(1(cons_0(1(1(y0)))))) -> c1(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(1(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(cons_0(y0)))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: 0'_1, ENCARG_1, ENCODE_0_1 Compound Symbols: c4, c1_2, c1_1, c5_2, c5_1, c5_3, c3_1, c_1 ---------------------------------------- (85) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(1(1(y0))) -> c(ENCARG(1(y0))) by ENCARG(1(1(1(y0)))) -> c(ENCARG(1(1(y0)))) ENCARG(1(1(cons_0(1(cons_0(y0)))))) -> c(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(1(1(cons_0(1(y0))))) -> c(ENCARG(1(cons_0(1(y0))))) ENCARG(1(1(cons_0(1(1(y0)))))) -> c(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(1(1(cons_0(cons_0(1(y0)))))) -> c(ENCARG(1(cons_0(cons_0(1(y0)))))) ENCARG(1(1(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(1(cons_0(cons_0(cons_0(y0)))))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4 ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) ENCODE_0(1(cons_0(z0))) -> c3(0'(1(0(encArg(z0))))) ENCODE_0(cons_0(1(z0))) -> c3(0'(0(1(encArg(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c3(0'(0(0(encArg(z0))))) ENCARG(1(cons_0(1(cons_0(y0))))) -> c(ENCARG(cons_0(1(cons_0(y0))))) ENCARG(1(cons_0(1(y0)))) -> c(ENCARG(cons_0(1(y0)))) ENCARG(1(cons_0(1(1(y0))))) -> c(ENCARG(cons_0(1(1(y0))))) ENCARG(1(cons_0(cons_0(1(y0))))) -> c(ENCARG(cons_0(cons_0(1(y0))))) ENCARG(1(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) ENCARG(cons_0(1(cons_0(1(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(cons_0(1(cons_0(1(y0))))) -> c1(ENCARG(1(cons_0(1(y0))))) ENCARG(cons_0(1(cons_0(1(1(y0)))))) -> c1(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(1(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(cons_0(y0)))))) ENCARG(1(1(1(y0)))) -> c(ENCARG(1(1(y0)))) ENCARG(1(1(cons_0(1(cons_0(y0)))))) -> c(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(1(1(cons_0(1(y0))))) -> c(ENCARG(1(cons_0(1(y0))))) ENCARG(1(1(cons_0(1(1(y0)))))) -> c(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(1(1(cons_0(cons_0(1(y0)))))) -> c(ENCARG(1(cons_0(cons_0(1(y0)))))) ENCARG(1(1(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(1(cons_0(cons_0(cons_0(y0)))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: 0'_1, ENCARG_1, ENCODE_0_1 Compound Symbols: c4, c1_2, c1_1, c5_2, c5_1, c5_3, c3_1, c_1 ---------------------------------------- (87) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) by ENCARG(cons_0(1(1(1(y0))))) -> c1(ENCARG(1(1(1(y0))))) ENCARG(cons_0(1(1(cons_0(1(cons_0(y0))))))) -> c1(ENCARG(1(1(cons_0(1(cons_0(y0))))))) ENCARG(cons_0(1(1(cons_0(1(y0)))))) -> c1(ENCARG(1(1(cons_0(1(y0)))))) ENCARG(cons_0(1(1(cons_0(1(1(y0))))))) -> c1(ENCARG(1(1(cons_0(1(1(y0))))))) ENCARG(cons_0(1(1(cons_0(cons_0(1(y0))))))) -> c1(ENCARG(1(1(cons_0(cons_0(1(y0))))))) ENCARG(cons_0(1(1(cons_0(cons_0(cons_0(y0))))))) -> c1(ENCARG(1(1(cons_0(cons_0(cons_0(y0))))))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4 ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) ENCODE_0(1(cons_0(z0))) -> c3(0'(1(0(encArg(z0))))) ENCODE_0(cons_0(1(z0))) -> c3(0'(0(1(encArg(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c3(0'(0(0(encArg(z0))))) ENCARG(1(cons_0(1(cons_0(y0))))) -> c(ENCARG(cons_0(1(cons_0(y0))))) ENCARG(1(cons_0(1(y0)))) -> c(ENCARG(cons_0(1(y0)))) ENCARG(1(cons_0(1(1(y0))))) -> c(ENCARG(cons_0(1(1(y0))))) ENCARG(1(cons_0(cons_0(1(y0))))) -> c(ENCARG(cons_0(cons_0(1(y0))))) ENCARG(1(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) ENCARG(cons_0(1(cons_0(1(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(cons_0(1(cons_0(1(y0))))) -> c1(ENCARG(1(cons_0(1(y0))))) ENCARG(cons_0(1(cons_0(1(1(y0)))))) -> c1(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(1(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(cons_0(y0)))))) ENCARG(1(1(1(y0)))) -> c(ENCARG(1(1(y0)))) ENCARG(1(1(cons_0(1(cons_0(y0)))))) -> c(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(1(1(cons_0(1(y0))))) -> c(ENCARG(1(cons_0(1(y0))))) ENCARG(1(1(cons_0(1(1(y0)))))) -> c(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(1(1(cons_0(cons_0(1(y0)))))) -> c(ENCARG(1(cons_0(cons_0(1(y0)))))) ENCARG(1(1(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(1(cons_0(cons_0(cons_0(y0)))))) ENCARG(cons_0(1(1(1(y0))))) -> c1(ENCARG(1(1(1(y0))))) ENCARG(cons_0(1(1(cons_0(1(cons_0(y0))))))) -> c1(ENCARG(1(1(cons_0(1(cons_0(y0))))))) ENCARG(cons_0(1(1(cons_0(1(y0)))))) -> c1(ENCARG(1(1(cons_0(1(y0)))))) ENCARG(cons_0(1(1(cons_0(1(1(y0))))))) -> c1(ENCARG(1(1(cons_0(1(1(y0))))))) ENCARG(cons_0(1(1(cons_0(cons_0(1(y0))))))) -> c1(ENCARG(1(1(cons_0(cons_0(1(y0))))))) ENCARG(cons_0(1(1(cons_0(cons_0(cons_0(y0))))))) -> c1(ENCARG(1(1(cons_0(cons_0(cons_0(y0))))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: 0'_1, ENCARG_1, ENCODE_0_1 Compound Symbols: c4, c1_2, c1_1, c5_2, c5_1, c5_3, c3_1, c_1 ---------------------------------------- (89) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(1(cons_0(1(y0)))) -> c(ENCARG(cons_0(1(y0)))) by ENCARG(1(cons_0(1(cons_0(y0))))) -> c(ENCARG(cons_0(1(cons_0(y0))))) ENCARG(1(cons_0(1(1(y0))))) -> c(ENCARG(cons_0(1(1(y0))))) ENCARG(1(cons_0(1(cons_0(1(cons_0(y0))))))) -> c(ENCARG(cons_0(1(cons_0(1(cons_0(y0))))))) ENCARG(1(cons_0(1(cons_0(1(y0)))))) -> c(ENCARG(cons_0(1(cons_0(1(y0)))))) ENCARG(1(cons_0(1(cons_0(1(1(y0))))))) -> c(ENCARG(cons_0(1(cons_0(1(1(y0))))))) ENCARG(1(cons_0(1(cons_0(cons_0(1(y0))))))) -> c(ENCARG(cons_0(1(cons_0(cons_0(1(y0))))))) ENCARG(1(cons_0(1(cons_0(cons_0(cons_0(y0))))))) -> c(ENCARG(cons_0(1(cons_0(cons_0(cons_0(y0))))))) ENCARG(1(cons_0(1(1(1(y0)))))) -> c(ENCARG(cons_0(1(1(1(y0)))))) ENCARG(1(cons_0(1(1(cons_0(1(cons_0(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(cons_0(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(1(y0))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(y0))))))) ENCARG(1(cons_0(1(1(cons_0(1(1(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(1(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(cons_0(1(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(cons_0(1(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(cons_0(cons_0(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(cons_0(cons_0(y0)))))))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4 ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(1(1(z0)))) -> c1(ENCARG(1(1(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) ENCODE_0(1(cons_0(z0))) -> c3(0'(1(0(encArg(z0))))) ENCODE_0(cons_0(1(z0))) -> c3(0'(0(1(encArg(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c3(0'(0(0(encArg(z0))))) ENCARG(1(cons_0(1(cons_0(y0))))) -> c(ENCARG(cons_0(1(cons_0(y0))))) ENCARG(1(cons_0(1(1(y0))))) -> c(ENCARG(cons_0(1(1(y0))))) ENCARG(1(cons_0(cons_0(1(y0))))) -> c(ENCARG(cons_0(cons_0(1(y0))))) ENCARG(1(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) ENCARG(cons_0(1(cons_0(1(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(cons_0(1(cons_0(1(y0))))) -> c1(ENCARG(1(cons_0(1(y0))))) ENCARG(cons_0(1(cons_0(1(1(y0)))))) -> c1(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(1(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(cons_0(y0)))))) ENCARG(1(1(1(y0)))) -> c(ENCARG(1(1(y0)))) ENCARG(1(1(cons_0(1(cons_0(y0)))))) -> c(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(1(1(cons_0(1(y0))))) -> c(ENCARG(1(cons_0(1(y0))))) ENCARG(1(1(cons_0(1(1(y0)))))) -> c(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(1(1(cons_0(cons_0(1(y0)))))) -> c(ENCARG(1(cons_0(cons_0(1(y0)))))) ENCARG(1(1(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(1(cons_0(cons_0(cons_0(y0)))))) ENCARG(cons_0(1(1(1(y0))))) -> c1(ENCARG(1(1(1(y0))))) ENCARG(cons_0(1(1(cons_0(1(cons_0(y0))))))) -> c1(ENCARG(1(1(cons_0(1(cons_0(y0))))))) ENCARG(cons_0(1(1(cons_0(1(y0)))))) -> c1(ENCARG(1(1(cons_0(1(y0)))))) ENCARG(cons_0(1(1(cons_0(1(1(y0))))))) -> c1(ENCARG(1(1(cons_0(1(1(y0))))))) ENCARG(cons_0(1(1(cons_0(cons_0(1(y0))))))) -> c1(ENCARG(1(1(cons_0(cons_0(1(y0))))))) ENCARG(cons_0(1(1(cons_0(cons_0(cons_0(y0))))))) -> c1(ENCARG(1(1(cons_0(cons_0(cons_0(y0))))))) ENCARG(1(cons_0(1(cons_0(1(cons_0(y0))))))) -> c(ENCARG(cons_0(1(cons_0(1(cons_0(y0))))))) ENCARG(1(cons_0(1(cons_0(1(y0)))))) -> c(ENCARG(cons_0(1(cons_0(1(y0)))))) ENCARG(1(cons_0(1(cons_0(1(1(y0))))))) -> c(ENCARG(cons_0(1(cons_0(1(1(y0))))))) ENCARG(1(cons_0(1(cons_0(cons_0(1(y0))))))) -> c(ENCARG(cons_0(1(cons_0(cons_0(1(y0))))))) ENCARG(1(cons_0(1(cons_0(cons_0(cons_0(y0))))))) -> c(ENCARG(cons_0(1(cons_0(cons_0(cons_0(y0))))))) ENCARG(1(cons_0(1(1(1(y0)))))) -> c(ENCARG(cons_0(1(1(1(y0)))))) ENCARG(1(cons_0(1(1(cons_0(1(cons_0(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(cons_0(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(1(y0))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(y0))))))) ENCARG(1(cons_0(1(1(cons_0(1(1(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(1(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(cons_0(1(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(cons_0(1(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(cons_0(cons_0(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(cons_0(cons_0(y0)))))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: 0'_1, ENCARG_1, ENCODE_0_1 Compound Symbols: c4, c1_2, c1_1, c5_2, c5_1, c5_3, c3_1, c_1 ---------------------------------------- (91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_0(1(1(y0)))) -> c1(ENCARG(1(1(y0)))) by ENCARG(cons_0(1(1(1(y0))))) -> c1(ENCARG(1(1(1(y0))))) ENCARG(cons_0(1(1(cons_0(1(cons_0(y0))))))) -> c1(ENCARG(1(1(cons_0(1(cons_0(y0))))))) ENCARG(cons_0(1(1(cons_0(1(y0)))))) -> c1(ENCARG(1(1(cons_0(1(y0)))))) ENCARG(cons_0(1(1(cons_0(1(1(y0))))))) -> c1(ENCARG(1(1(cons_0(1(1(y0))))))) ENCARG(cons_0(1(1(cons_0(cons_0(1(y0))))))) -> c1(ENCARG(1(1(cons_0(cons_0(1(y0))))))) ENCARG(cons_0(1(1(cons_0(cons_0(cons_0(y0))))))) -> c1(ENCARG(1(1(cons_0(cons_0(cons_0(y0))))))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4 ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) ENCODE_0(1(cons_0(z0))) -> c3(0'(1(0(encArg(z0))))) ENCODE_0(cons_0(1(z0))) -> c3(0'(0(1(encArg(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c3(0'(0(0(encArg(z0))))) ENCARG(1(cons_0(1(cons_0(y0))))) -> c(ENCARG(cons_0(1(cons_0(y0))))) ENCARG(1(cons_0(1(1(y0))))) -> c(ENCARG(cons_0(1(1(y0))))) ENCARG(1(cons_0(cons_0(1(y0))))) -> c(ENCARG(cons_0(cons_0(1(y0))))) ENCARG(1(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) ENCARG(cons_0(1(cons_0(1(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(cons_0(1(cons_0(1(y0))))) -> c1(ENCARG(1(cons_0(1(y0))))) ENCARG(cons_0(1(cons_0(1(1(y0)))))) -> c1(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(1(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(cons_0(y0)))))) ENCARG(1(1(1(y0)))) -> c(ENCARG(1(1(y0)))) ENCARG(1(1(cons_0(1(cons_0(y0)))))) -> c(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(1(1(cons_0(1(y0))))) -> c(ENCARG(1(cons_0(1(y0))))) ENCARG(1(1(cons_0(1(1(y0)))))) -> c(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(1(1(cons_0(cons_0(1(y0)))))) -> c(ENCARG(1(cons_0(cons_0(1(y0)))))) ENCARG(1(1(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(1(cons_0(cons_0(cons_0(y0)))))) ENCARG(cons_0(1(1(1(y0))))) -> c1(ENCARG(1(1(1(y0))))) ENCARG(cons_0(1(1(cons_0(1(cons_0(y0))))))) -> c1(ENCARG(1(1(cons_0(1(cons_0(y0))))))) ENCARG(cons_0(1(1(cons_0(1(y0)))))) -> c1(ENCARG(1(1(cons_0(1(y0)))))) ENCARG(cons_0(1(1(cons_0(1(1(y0))))))) -> c1(ENCARG(1(1(cons_0(1(1(y0))))))) ENCARG(cons_0(1(1(cons_0(cons_0(1(y0))))))) -> c1(ENCARG(1(1(cons_0(cons_0(1(y0))))))) ENCARG(cons_0(1(1(cons_0(cons_0(cons_0(y0))))))) -> c1(ENCARG(1(1(cons_0(cons_0(cons_0(y0))))))) ENCARG(1(cons_0(1(cons_0(1(cons_0(y0))))))) -> c(ENCARG(cons_0(1(cons_0(1(cons_0(y0))))))) ENCARG(1(cons_0(1(cons_0(1(y0)))))) -> c(ENCARG(cons_0(1(cons_0(1(y0)))))) ENCARG(1(cons_0(1(cons_0(1(1(y0))))))) -> c(ENCARG(cons_0(1(cons_0(1(1(y0))))))) ENCARG(1(cons_0(1(cons_0(cons_0(1(y0))))))) -> c(ENCARG(cons_0(1(cons_0(cons_0(1(y0))))))) ENCARG(1(cons_0(1(cons_0(cons_0(cons_0(y0))))))) -> c(ENCARG(cons_0(1(cons_0(cons_0(cons_0(y0))))))) ENCARG(1(cons_0(1(1(1(y0)))))) -> c(ENCARG(cons_0(1(1(1(y0)))))) ENCARG(1(cons_0(1(1(cons_0(1(cons_0(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(cons_0(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(1(y0))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(y0))))))) ENCARG(1(cons_0(1(1(cons_0(1(1(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(1(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(cons_0(1(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(cons_0(1(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(cons_0(cons_0(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(cons_0(cons_0(y0)))))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: 0'_1, ENCARG_1, ENCODE_0_1 Compound Symbols: c4, c1_2, c5_2, c5_1, c5_3, c3_1, c_1, c1_1 ---------------------------------------- (93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(1(cons_0(1(1(y0))))) -> c(ENCARG(cons_0(1(1(y0))))) by ENCARG(1(cons_0(1(1(1(y0)))))) -> c(ENCARG(cons_0(1(1(1(y0)))))) ENCARG(1(cons_0(1(1(cons_0(1(cons_0(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(cons_0(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(1(y0))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(y0))))))) ENCARG(1(cons_0(1(1(cons_0(1(1(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(1(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(cons_0(1(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(cons_0(1(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(cons_0(cons_0(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(cons_0(cons_0(y0)))))))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4 ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) ENCODE_0(1(cons_0(z0))) -> c3(0'(1(0(encArg(z0))))) ENCODE_0(cons_0(1(z0))) -> c3(0'(0(1(encArg(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c3(0'(0(0(encArg(z0))))) ENCARG(1(cons_0(1(cons_0(y0))))) -> c(ENCARG(cons_0(1(cons_0(y0))))) ENCARG(1(cons_0(1(1(y0))))) -> c(ENCARG(cons_0(1(1(y0))))) ENCARG(1(cons_0(cons_0(1(y0))))) -> c(ENCARG(cons_0(cons_0(1(y0))))) ENCARG(1(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) ENCARG(cons_0(1(cons_0(1(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(cons_0(1(cons_0(1(y0))))) -> c1(ENCARG(1(cons_0(1(y0))))) ENCARG(cons_0(1(cons_0(1(1(y0)))))) -> c1(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(1(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(cons_0(y0)))))) ENCARG(1(1(1(y0)))) -> c(ENCARG(1(1(y0)))) ENCARG(1(1(cons_0(1(cons_0(y0)))))) -> c(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(1(1(cons_0(1(y0))))) -> c(ENCARG(1(cons_0(1(y0))))) ENCARG(1(1(cons_0(1(1(y0)))))) -> c(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(1(1(cons_0(cons_0(1(y0)))))) -> c(ENCARG(1(cons_0(cons_0(1(y0)))))) ENCARG(1(1(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(1(cons_0(cons_0(cons_0(y0)))))) ENCARG(cons_0(1(1(1(y0))))) -> c1(ENCARG(1(1(1(y0))))) ENCARG(cons_0(1(1(cons_0(1(cons_0(y0))))))) -> c1(ENCARG(1(1(cons_0(1(cons_0(y0))))))) ENCARG(cons_0(1(1(cons_0(1(y0)))))) -> c1(ENCARG(1(1(cons_0(1(y0)))))) ENCARG(cons_0(1(1(cons_0(1(1(y0))))))) -> c1(ENCARG(1(1(cons_0(1(1(y0))))))) ENCARG(cons_0(1(1(cons_0(cons_0(1(y0))))))) -> c1(ENCARG(1(1(cons_0(cons_0(1(y0))))))) ENCARG(cons_0(1(1(cons_0(cons_0(cons_0(y0))))))) -> c1(ENCARG(1(1(cons_0(cons_0(cons_0(y0))))))) ENCARG(1(cons_0(1(cons_0(1(cons_0(y0))))))) -> c(ENCARG(cons_0(1(cons_0(1(cons_0(y0))))))) ENCARG(1(cons_0(1(cons_0(1(y0)))))) -> c(ENCARG(cons_0(1(cons_0(1(y0)))))) ENCARG(1(cons_0(1(cons_0(1(1(y0))))))) -> c(ENCARG(cons_0(1(cons_0(1(1(y0))))))) ENCARG(1(cons_0(1(cons_0(cons_0(1(y0))))))) -> c(ENCARG(cons_0(1(cons_0(cons_0(1(y0))))))) ENCARG(1(cons_0(1(cons_0(cons_0(cons_0(y0))))))) -> c(ENCARG(cons_0(1(cons_0(cons_0(cons_0(y0))))))) ENCARG(1(cons_0(1(1(1(y0)))))) -> c(ENCARG(cons_0(1(1(1(y0)))))) ENCARG(1(cons_0(1(1(cons_0(1(cons_0(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(cons_0(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(1(y0))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(y0))))))) ENCARG(1(cons_0(1(1(cons_0(1(1(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(1(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(cons_0(1(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(cons_0(1(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(cons_0(cons_0(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(cons_0(cons_0(y0)))))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: 0'_1, ENCARG_1, ENCODE_0_1 Compound Symbols: c4, c1_2, c5_2, c5_1, c5_3, c3_1, c_1, c1_1 ---------------------------------------- (95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_0(1(cons_0(1(y0))))) -> c1(ENCARG(1(cons_0(1(y0))))) by ENCARG(cons_0(1(cons_0(1(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(cons_0(1(cons_0(1(1(y0)))))) -> c1(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(cons_0(1(cons_0(1(cons_0(1(cons_0(y0)))))))) -> c1(ENCARG(1(cons_0(1(cons_0(1(cons_0(y0)))))))) ENCARG(cons_0(1(cons_0(1(cons_0(1(y0))))))) -> c1(ENCARG(1(cons_0(1(cons_0(1(y0))))))) ENCARG(cons_0(1(cons_0(1(cons_0(1(1(y0)))))))) -> c1(ENCARG(1(cons_0(1(cons_0(1(1(y0)))))))) ENCARG(cons_0(1(cons_0(1(cons_0(cons_0(1(y0)))))))) -> c1(ENCARG(1(cons_0(1(cons_0(cons_0(1(y0)))))))) ENCARG(cons_0(1(cons_0(1(cons_0(cons_0(cons_0(y0)))))))) -> c1(ENCARG(1(cons_0(1(cons_0(cons_0(cons_0(y0)))))))) ENCARG(cons_0(1(cons_0(1(1(1(y0))))))) -> c1(ENCARG(1(cons_0(1(1(1(y0))))))) ENCARG(cons_0(1(cons_0(1(1(cons_0(1(cons_0(y0))))))))) -> c1(ENCARG(1(cons_0(1(1(cons_0(1(cons_0(y0))))))))) ENCARG(cons_0(1(cons_0(1(1(cons_0(1(y0)))))))) -> c1(ENCARG(1(cons_0(1(1(cons_0(1(y0)))))))) ENCARG(cons_0(1(cons_0(1(1(cons_0(1(1(y0))))))))) -> c1(ENCARG(1(cons_0(1(1(cons_0(1(1(y0))))))))) ENCARG(cons_0(1(cons_0(1(1(cons_0(cons_0(1(y0))))))))) -> c1(ENCARG(1(cons_0(1(1(cons_0(cons_0(1(y0))))))))) ENCARG(cons_0(1(cons_0(1(1(cons_0(cons_0(cons_0(y0))))))))) -> c1(ENCARG(1(cons_0(1(1(cons_0(cons_0(cons_0(y0))))))))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: encArg(1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) 0(0(0(0(z0)))) -> 1(0(1(1(z0)))) 0(1(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4 ENCARG(cons_0(1(cons_0(z0)))) -> c1(0'(1(0(encArg(z0)))), ENCARG(1(cons_0(z0)))) ENCARG(cons_0(cons_0(1(z0)))) -> c1(0'(0(1(encArg(z0)))), ENCARG(cons_0(1(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c1(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) ENCODE_0(1(cons_0(z0))) -> c3(0'(1(0(encArg(z0))))) ENCODE_0(cons_0(1(z0))) -> c3(0'(0(1(encArg(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c3(0'(0(0(encArg(z0))))) ENCARG(1(cons_0(1(cons_0(y0))))) -> c(ENCARG(cons_0(1(cons_0(y0))))) ENCARG(1(cons_0(1(1(y0))))) -> c(ENCARG(cons_0(1(1(y0))))) ENCARG(1(cons_0(cons_0(1(y0))))) -> c(ENCARG(cons_0(cons_0(1(y0))))) ENCARG(1(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) ENCARG(cons_0(1(cons_0(1(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(cons_0(1(cons_0(1(1(y0)))))) -> c1(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(1(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(1(y0)))))) ENCARG(cons_0(1(cons_0(cons_0(cons_0(y0)))))) -> c1(ENCARG(1(cons_0(cons_0(cons_0(y0)))))) ENCARG(1(1(1(y0)))) -> c(ENCARG(1(1(y0)))) ENCARG(1(1(cons_0(1(cons_0(y0)))))) -> c(ENCARG(1(cons_0(1(cons_0(y0)))))) ENCARG(1(1(cons_0(1(y0))))) -> c(ENCARG(1(cons_0(1(y0))))) ENCARG(1(1(cons_0(1(1(y0)))))) -> c(ENCARG(1(cons_0(1(1(y0)))))) ENCARG(1(1(cons_0(cons_0(1(y0)))))) -> c(ENCARG(1(cons_0(cons_0(1(y0)))))) ENCARG(1(1(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(1(cons_0(cons_0(cons_0(y0)))))) ENCARG(cons_0(1(1(1(y0))))) -> c1(ENCARG(1(1(1(y0))))) ENCARG(cons_0(1(1(cons_0(1(cons_0(y0))))))) -> c1(ENCARG(1(1(cons_0(1(cons_0(y0))))))) ENCARG(cons_0(1(1(cons_0(1(y0)))))) -> c1(ENCARG(1(1(cons_0(1(y0)))))) ENCARG(cons_0(1(1(cons_0(1(1(y0))))))) -> c1(ENCARG(1(1(cons_0(1(1(y0))))))) ENCARG(cons_0(1(1(cons_0(cons_0(1(y0))))))) -> c1(ENCARG(1(1(cons_0(cons_0(1(y0))))))) ENCARG(cons_0(1(1(cons_0(cons_0(cons_0(y0))))))) -> c1(ENCARG(1(1(cons_0(cons_0(cons_0(y0))))))) ENCARG(1(cons_0(1(cons_0(1(cons_0(y0))))))) -> c(ENCARG(cons_0(1(cons_0(1(cons_0(y0))))))) ENCARG(1(cons_0(1(cons_0(1(y0)))))) -> c(ENCARG(cons_0(1(cons_0(1(y0)))))) ENCARG(1(cons_0(1(cons_0(1(1(y0))))))) -> c(ENCARG(cons_0(1(cons_0(1(1(y0))))))) ENCARG(1(cons_0(1(cons_0(cons_0(1(y0))))))) -> c(ENCARG(cons_0(1(cons_0(cons_0(1(y0))))))) ENCARG(1(cons_0(1(cons_0(cons_0(cons_0(y0))))))) -> c(ENCARG(cons_0(1(cons_0(cons_0(cons_0(y0))))))) ENCARG(1(cons_0(1(1(1(y0)))))) -> c(ENCARG(cons_0(1(1(1(y0)))))) ENCARG(1(cons_0(1(1(cons_0(1(cons_0(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(cons_0(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(1(y0))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(y0))))))) ENCARG(1(cons_0(1(1(cons_0(1(1(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(1(1(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(cons_0(1(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(cons_0(1(y0)))))))) ENCARG(1(cons_0(1(1(cons_0(cons_0(cons_0(y0)))))))) -> c(ENCARG(cons_0(1(1(cons_0(cons_0(cons_0(y0)))))))) ENCARG(cons_0(1(cons_0(1(cons_0(1(cons_0(y0)))))))) -> c1(ENCARG(1(cons_0(1(cons_0(1(cons_0(y0)))))))) ENCARG(cons_0(1(cons_0(1(cons_0(1(y0))))))) -> c1(ENCARG(1(cons_0(1(cons_0(1(y0))))))) ENCARG(cons_0(1(cons_0(1(cons_0(1(1(y0)))))))) -> c1(ENCARG(1(cons_0(1(cons_0(1(1(y0)))))))) ENCARG(cons_0(1(cons_0(1(cons_0(cons_0(1(y0)))))))) -> c1(ENCARG(1(cons_0(1(cons_0(cons_0(1(y0)))))))) ENCARG(cons_0(1(cons_0(1(cons_0(cons_0(cons_0(y0)))))))) -> c1(ENCARG(1(cons_0(1(cons_0(cons_0(cons_0(y0)))))))) ENCARG(cons_0(1(cons_0(1(1(1(y0))))))) -> c1(ENCARG(1(cons_0(1(1(1(y0))))))) ENCARG(cons_0(1(cons_0(1(1(cons_0(1(cons_0(y0))))))))) -> c1(ENCARG(1(cons_0(1(1(cons_0(1(cons_0(y0))))))))) ENCARG(cons_0(1(cons_0(1(1(cons_0(1(y0)))))))) -> c1(ENCARG(1(cons_0(1(1(cons_0(1(y0)))))))) ENCARG(cons_0(1(cons_0(1(1(cons_0(1(1(y0))))))))) -> c1(ENCARG(1(cons_0(1(1(cons_0(1(1(y0))))))))) ENCARG(cons_0(1(cons_0(1(1(cons_0(cons_0(1(y0))))))))) -> c1(ENCARG(1(cons_0(1(1(cons_0(cons_0(1(y0))))))))) ENCARG(cons_0(1(cons_0(1(1(cons_0(cons_0(cons_0(y0))))))))) -> c1(ENCARG(1(cons_0(1(1(cons_0(cons_0(cons_0(y0))))))))) S tuples: 0'(0(0(0(z0)))) -> c4 0'(1(0(1(1(x0))))) -> c5(0'(1(0(1(x0)))), 0'(1(x0))) 0'(1(0(1(1(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(0(0(z0)))))))) -> c5(0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(1(1(0(1(z0)))))))) -> c5(0'(0(0(1(0(0(1(0(z0)))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(1(z0)))))))) -> c5(0'(0(1(0(0(0(1(0(z0)))))))), 0'(1(0(1(0(1(1(z0))))))), 0'(1(0(1(1(z0)))))) 0'(1(0(1(1(0(1(1(0(1(z0)))))))))) -> c5(0'(0(1(0(0(1(0(0(1(0(z0)))))))))), 0'(1(0(1(0(1(1(0(1(z0))))))))), 0'(1(0(1(1(0(1(z0)))))))) 0'(1(0(1(1(0(1(x0))))))) -> c5(0'(1(0(1(0(1(x0))))))) 0'(1(0(1(1(0(1(0(0(0(z0)))))))))) -> c5(0'(1(0(1(0(1(0(0(0(z0))))))))), 0'(1(0(1(0(0(0(z0)))))))) 0'(1(0(1(0(0(0(z0))))))) -> c5(0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1 Defined Pair Symbols: 0'_1, ENCARG_1, ENCODE_0_1 Compound Symbols: c4, c1_2, c5_2, c5_1, c5_3, c3_1, c_1, c1_1