/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^2)) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^2). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 48 ms] (4) CpxRelTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (6) CdtProblem (7) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CdtProblem (9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (12) CdtProblem (13) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CdtProblem (15) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CdtProblem (17) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CdtProblem (19) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CdtProblem (21) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CdtProblem (23) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 15 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms] (58) CdtProblem (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 15 ms] (64) CdtProblem (65) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 10.9 s] (88) CdtProblem (89) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (90) BOUNDS(1, 1) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: 0(0(0(0(x1)))) -> 0(0(0(1(x1)))) 1(0(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(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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, n^2). The TRS R consists of the following rules: 0(0(0(0(x1)))) -> 0(0(0(1(x1)))) 1(0(0(1(x1)))) -> 0(0(1(0(x1)))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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, n^2). The TRS R consists of the following rules: 0(0(0(0(x1)))) -> 0(0(0(1(x1)))) 1(0(0(1(x1)))) -> 0(0(1(0(x1)))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) encode_0(z0) -> 0(encArg(z0)) encode_1(z0) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_0(z0)) -> c(0'(encArg(z0)), ENCARG(z0)) ENCARG(cons_1(z0)) -> c1(1'(encArg(z0)), ENCARG(z0)) ENCODE_0(z0) -> c2(0'(encArg(z0)), ENCARG(z0)) ENCODE_1(z0) -> c3(1'(encArg(z0)), ENCARG(z0)) 0'(0(0(0(z0)))) -> c4(0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)), 1'(z0)) 1'(0(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 1'(0(z0)), 0'(z0)) S tuples: 0'(0(0(0(z0)))) -> c4(0'(0(0(1(z0)))), 0'(0(1(z0))), 0'(1(z0)), 1'(z0)) 1'(0(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 1'(0(z0)), 0'(z0)) K tuples:none Defined Rule Symbols: 0_1, 1_1, encArg_1, encode_0_1, encode_1_1 Defined Pair Symbols: ENCARG_1, ENCODE_0_1, ENCODE_1_1, 0'_1, 1'_1 Compound Symbols: c_2, c1_2, c2_2, c3_2, c4_4, c5_4 ---------------------------------------- (7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) encode_0(z0) -> 0(encArg(z0)) encode_1(z0) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_0(z0)) -> c(0'(encArg(z0)), ENCARG(z0)) ENCARG(cons_1(z0)) -> c1(1'(encArg(z0)), ENCARG(z0)) ENCODE_0(z0) -> c2(0'(encArg(z0)), ENCARG(z0)) ENCODE_1(z0) -> c3(1'(encArg(z0)), ENCARG(z0)) 1'(0(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 1'(0(z0)), 0'(z0)) 0'(0(0(0(z0)))) -> c4(1'(z0)) S tuples: 1'(0(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 1'(0(z0)), 0'(z0)) 0'(0(0(0(z0)))) -> c4(1'(z0)) K tuples:none Defined Rule Symbols: 0_1, 1_1, encArg_1, encode_0_1, encode_1_1 Defined Pair Symbols: ENCARG_1, ENCODE_0_1, ENCODE_1_1, 1'_1, 0'_1 Compound Symbols: c_2, c1_2, c2_2, c3_2, c5_4, c4_1 ---------------------------------------- (9) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) encode_0(z0) -> 0(encArg(z0)) encode_1(z0) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_0(z0)) -> c(0'(encArg(z0)), ENCARG(z0)) ENCARG(cons_1(z0)) -> c1(1'(encArg(z0)), ENCARG(z0)) 1'(0(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 1'(0(z0)), 0'(z0)) 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCODE_0(z0) -> c6(0'(encArg(z0))) ENCODE_0(z0) -> c6(ENCARG(z0)) ENCODE_1(z0) -> c6(1'(encArg(z0))) ENCODE_1(z0) -> c6(ENCARG(z0)) S tuples: 1'(0(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 1'(0(z0)), 0'(z0)) 0'(0(0(0(z0)))) -> c4(1'(z0)) K tuples:none Defined Rule Symbols: 0_1, 1_1, encArg_1, encode_0_1, encode_1_1 Defined Pair Symbols: ENCARG_1, 1'_1, 0'_1, ENCODE_0_1, ENCODE_1_1 Compound Symbols: c_2, c1_2, c5_4, c4_1, c6_1 ---------------------------------------- (11) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ENCODE_0(z0) -> c6(ENCARG(z0)) ENCODE_1(z0) -> c6(ENCARG(z0)) ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) encode_0(z0) -> 0(encArg(z0)) encode_1(z0) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_0(z0)) -> c(0'(encArg(z0)), ENCARG(z0)) ENCARG(cons_1(z0)) -> c1(1'(encArg(z0)), ENCARG(z0)) 1'(0(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 1'(0(z0)), 0'(z0)) 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCODE_0(z0) -> c6(0'(encArg(z0))) ENCODE_1(z0) -> c6(1'(encArg(z0))) S tuples: 1'(0(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 1'(0(z0)), 0'(z0)) 0'(0(0(0(z0)))) -> c4(1'(z0)) K tuples:none Defined Rule Symbols: 0_1, 1_1, encArg_1, encode_0_1, encode_1_1 Defined Pair Symbols: ENCARG_1, 1'_1, 0'_1, ENCODE_0_1, ENCODE_1_1 Compound Symbols: c_2, c1_2, c5_4, c4_1, c6_1 ---------------------------------------- (13) 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)) ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_0(z0)) -> c(0'(encArg(z0)), ENCARG(z0)) ENCARG(cons_1(z0)) -> c1(1'(encArg(z0)), ENCARG(z0)) 1'(0(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 1'(0(z0)), 0'(z0)) 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCODE_0(z0) -> c6(0'(encArg(z0))) ENCODE_1(z0) -> c6(1'(encArg(z0))) S tuples: 1'(0(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 1'(0(z0)), 0'(z0)) 0'(0(0(0(z0)))) -> c4(1'(z0)) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, 1'_1, 0'_1, ENCODE_0_1, ENCODE_1_1 Compound Symbols: c_2, c1_2, c5_4, c4_1, c6_1 ---------------------------------------- (15) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_0(z0)) -> c(0'(encArg(z0)), ENCARG(z0)) by ENCARG(cons_0(cons_0(z0))) -> c(0'(0(encArg(z0))), ENCARG(cons_0(z0))) ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0))) ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(z0)) -> c1(1'(encArg(z0)), ENCARG(z0)) 1'(0(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 1'(0(z0)), 0'(z0)) 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCODE_0(z0) -> c6(0'(encArg(z0))) ENCODE_1(z0) -> c6(1'(encArg(z0))) ENCARG(cons_0(cons_0(z0))) -> c(0'(0(encArg(z0))), ENCARG(cons_0(z0))) ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0))) S tuples: 1'(0(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 1'(0(z0)), 0'(z0)) 0'(0(0(0(z0)))) -> c4(1'(z0)) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, 1'_1, 0'_1, ENCODE_0_1, ENCODE_1_1 Compound Symbols: c1_2, c5_4, c4_1, c6_1, c_2 ---------------------------------------- (17) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_1(z0)) -> c1(1'(encArg(z0)), ENCARG(z0)) by ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0))) ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0))) ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 1'(0(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 1'(0(z0)), 0'(z0)) 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCODE_0(z0) -> c6(0'(encArg(z0))) ENCODE_1(z0) -> c6(1'(encArg(z0))) ENCARG(cons_0(cons_0(z0))) -> c(0'(0(encArg(z0))), ENCARG(cons_0(z0))) ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0))) ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0))) ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0))) S tuples: 1'(0(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 1'(0(z0)), 0'(z0)) 0'(0(0(0(z0)))) -> c4(1'(z0)) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 1'_1, 0'_1, ENCODE_0_1, ENCODE_1_1, ENCARG_1 Compound Symbols: c5_4, c4_1, c6_1, c_2, c1_2 ---------------------------------------- (19) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace 1'(0(0(1(z0)))) -> c5(0'(0(1(0(z0)))), 0'(1(0(z0))), 1'(0(z0)), 0'(z0)) by 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0)))), 0'(0(1(z0)))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(0(1(0(0(0(1(z0))))))), 0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCODE_0(z0) -> c6(0'(encArg(z0))) ENCODE_1(z0) -> c6(1'(encArg(z0))) ENCARG(cons_0(cons_0(z0))) -> c(0'(0(encArg(z0))), ENCARG(cons_0(z0))) ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0))) ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0))) ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0)))), 0'(0(1(z0)))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(0(1(0(0(0(1(z0))))))), 0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) S tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0)))), 0'(0(1(z0)))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(0(1(0(0(0(1(z0))))))), 0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 0'_1, ENCODE_0_1, ENCODE_1_1, ENCARG_1, 1'_1 Compound Symbols: c4_1, c6_1, c_2, c1_2, c5_4 ---------------------------------------- (21) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCODE_0(z0) -> c6(0'(encArg(z0))) ENCODE_1(z0) -> c6(1'(encArg(z0))) ENCARG(cons_0(cons_0(z0))) -> c(0'(0(encArg(z0))), ENCARG(cons_0(z0))) ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0))) ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0))) ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) S tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 0'_1, ENCODE_0_1, ENCODE_1_1, ENCARG_1, 1'_1 Compound Symbols: c4_1, c6_1, c_2, c1_2, c5_3 ---------------------------------------- (23) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_0(z0) -> c6(0'(encArg(z0))) by ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0)))) ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0)))) ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCODE_1(z0) -> c6(1'(encArg(z0))) ENCARG(cons_0(cons_0(z0))) -> c(0'(0(encArg(z0))), ENCARG(cons_0(z0))) ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0))) ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0))) ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0)))) ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0)))) S tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 0'_1, ENCODE_1_1, ENCARG_1, 1'_1, ENCODE_0_1 Compound Symbols: c4_1, c6_1, c_2, c1_2, c5_3 ---------------------------------------- (25) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_1(z0) -> c6(1'(encArg(z0))) by ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0)))) ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0)))) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCARG(cons_0(cons_0(z0))) -> c(0'(0(encArg(z0))), ENCARG(cons_0(z0))) ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0))) ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0))) ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0)))) ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0)))) ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0)))) ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0)))) S tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 0'_1, ENCARG_1, 1'_1, ENCODE_0_1, ENCODE_1_1 Compound Symbols: c4_1, c_2, c1_2, c5_3, c6_1 ---------------------------------------- (27) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_0(cons_0(z0))) -> c(0'(0(encArg(z0))), ENCARG(cons_0(z0))) by ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0))) ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0))) ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0)))) ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0)))) ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0)))) ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) S tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 0'_1, ENCARG_1, 1'_1, ENCODE_0_1, ENCODE_1_1 Compound Symbols: c4_1, c_2, c1_2, c5_3, c6_1 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_0(cons_1(z0))) -> c(0'(1(encArg(z0))), ENCARG(cons_1(z0))) by ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0))) ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0)))) ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0)))) ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0)))) ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) S tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 0'_1, ENCARG_1, 1'_1, ENCODE_0_1, ENCODE_1_1 Compound Symbols: c4_1, c1_2, c5_3, c6_1, c_2 ---------------------------------------- (31) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_1(cons_0(z0))) -> c1(1'(0(encArg(z0))), ENCARG(cons_0(z0))) by ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0)))) ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0)))) ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0)))) ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) S tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 0'_1, ENCARG_1, 1'_1, ENCODE_0_1, ENCODE_1_1 Compound Symbols: c4_1, c1_2, c5_3, c6_1, c_2 ---------------------------------------- (33) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_1(cons_1(z0))) -> c1(1'(1(encArg(z0))), ENCARG(cons_1(z0))) by ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0)))) ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0)))) ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0)))) ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) S tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 0'_1, 1'_1, ENCODE_0_1, ENCODE_1_1, ENCARG_1 Compound Symbols: c4_1, c5_3, c6_1, c_2, c1_2 ---------------------------------------- (35) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(1(0(0(1(z0))))), 1'(0(0(1(z0))))) by 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0)))) ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0)))) ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0)))) ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) S tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 0'_1, 1'_1, ENCODE_0_1, ENCODE_1_1, ENCARG_1 Compound Symbols: c4_1, c5_3, c6_1, c_2, c1_2, c5_1 ---------------------------------------- (37) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(0(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) by 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(1(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0)))) ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0)))) ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0)))) ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(1(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) S tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(0'(1(0(0(0(1(z0)))))), 1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 0'_1, ENCODE_0_1, ENCODE_1_1, ENCARG_1, 1'_1 Compound Symbols: c4_1, c6_1, c_2, c1_2, c5_3, c5_1 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0)))) ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0)))) ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0)))) ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) S tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 0'_1, ENCODE_0_1, ENCODE_1_1, ENCARG_1, 1'_1 Compound Symbols: c4_1, c6_1, c_2, c1_2, c5_3, c5_1, c5_2 ---------------------------------------- (41) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_0(cons_0(z0)) -> c6(0'(0(encArg(z0)))) by ENCODE_0(cons_0(cons_0(z0))) -> c6(0'(0(0(encArg(z0))))) ENCODE_0(cons_0(cons_1(z0))) -> c6(0'(0(1(encArg(z0))))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0)))) ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0)))) ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c6(0'(0(0(encArg(z0))))) ENCODE_0(cons_0(cons_1(z0))) -> c6(0'(0(1(encArg(z0))))) S tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 0'_1, ENCODE_0_1, ENCODE_1_1, ENCARG_1, 1'_1 Compound Symbols: c4_1, c6_1, c_2, c1_2, c5_3, c5_1, c5_2 ---------------------------------------- (43) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_0(cons_1(z0)) -> c6(0'(1(encArg(z0)))) by ENCODE_0(cons_1(cons_0(z0))) -> c6(0'(1(0(encArg(z0))))) ENCODE_0(cons_1(cons_1(z0))) -> c6(0'(1(1(encArg(z0))))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0)))) ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c6(0'(0(0(encArg(z0))))) ENCODE_0(cons_0(cons_1(z0))) -> c6(0'(0(1(encArg(z0))))) ENCODE_0(cons_1(cons_0(z0))) -> c6(0'(1(0(encArg(z0))))) ENCODE_0(cons_1(cons_1(z0))) -> c6(0'(1(1(encArg(z0))))) S tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 0'_1, ENCODE_1_1, ENCARG_1, 1'_1, ENCODE_0_1 Compound Symbols: c4_1, c6_1, c_2, c1_2, c5_3, c5_1, c5_2 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_1(cons_0(z0)) -> c6(1'(0(encArg(z0)))) by ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0)))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c6(0'(0(0(encArg(z0))))) ENCODE_0(cons_0(cons_1(z0))) -> c6(0'(0(1(encArg(z0))))) ENCODE_0(cons_1(cons_0(z0))) -> c6(0'(1(0(encArg(z0))))) ENCODE_0(cons_1(cons_1(z0))) -> c6(0'(1(1(encArg(z0))))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) S tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 0'_1, ENCODE_1_1, ENCARG_1, 1'_1, ENCODE_0_1 Compound Symbols: c4_1, c6_1, c_2, c1_2, c5_3, c5_1, c5_2 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_1(cons_1(z0)) -> c6(1'(1(encArg(z0)))) by ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c6(0'(0(0(encArg(z0))))) ENCODE_0(cons_0(cons_1(z0))) -> c6(0'(0(1(encArg(z0))))) ENCODE_0(cons_1(cons_0(z0))) -> c6(0'(1(0(encArg(z0))))) ENCODE_0(cons_1(cons_1(z0))) -> c6(0'(1(1(encArg(z0))))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) S tuples: 0'(0(0(0(z0)))) -> c4(1'(z0)) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: 0'_1, ENCARG_1, 1'_1, ENCODE_0_1, ENCODE_1_1 Compound Symbols: c4_1, c_2, c1_2, c5_3, c5_1, c5_2, c6_1 ---------------------------------------- (49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace 0'(0(0(0(z0)))) -> c4(1'(z0)) by 0'(0(0(0(0(0(1(0(1(y0))))))))) -> c4(1'(0(0(1(0(1(y0))))))) 0'(0(0(0(0(0(1(0(0(0(y0)))))))))) -> c4(1'(0(0(1(0(0(0(y0)))))))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c6(0'(0(0(encArg(z0))))) ENCODE_0(cons_0(cons_1(z0))) -> c6(0'(0(1(encArg(z0))))) ENCODE_0(cons_1(cons_0(z0))) -> c6(0'(1(0(encArg(z0))))) ENCODE_0(cons_1(cons_1(z0))) -> c6(0'(1(1(encArg(z0))))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) 0'(0(0(0(0(0(1(0(1(y0))))))))) -> c4(1'(0(0(1(0(1(y0))))))) 0'(0(0(0(0(0(1(0(0(0(y0)))))))))) -> c4(1'(0(0(1(0(0(0(y0)))))))) S tuples: 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) 0'(0(0(0(0(0(1(0(1(y0))))))))) -> c4(1'(0(0(1(0(1(y0))))))) 0'(0(0(0(0(0(1(0(0(0(y0)))))))))) -> c4(1'(0(0(1(0(0(0(y0)))))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, 1'_1, ENCODE_0_1, ENCODE_1_1, 0'_1 Compound Symbols: c_2, c1_2, c5_3, c5_1, c5_2, c6_1, c4_1 ---------------------------------------- (51) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: 0'(0(0(0(0(0(1(0(0(0(y0)))))))))) -> c4(1'(0(0(1(0(0(0(y0)))))))) 0'(0(0(0(0(0(1(0(1(y0))))))))) -> c4(1'(0(0(1(0(1(y0))))))) Removed 4 trailing nodes: ENCODE_0(cons_1(cons_1(z0))) -> c6(0'(1(1(encArg(z0))))) ENCODE_0(cons_0(cons_1(z0))) -> c6(0'(0(1(encArg(z0))))) ENCODE_0(cons_1(cons_0(z0))) -> c6(0'(1(0(encArg(z0))))) ENCODE_0(cons_0(cons_0(z0))) -> c6(0'(0(0(encArg(z0))))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(0'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(0'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(0'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(0'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) S tuples: 1'(0(0(1(0(1(z0)))))) -> c5(0'(0(0(0(1(0(z0)))))), 0'(0(0(1(0(z0))))), 1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5(0'(0(0(0(1(0(x0))))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0))))), 0'(0(0(0(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, 1'_1, ENCODE_1_1 Compound Symbols: c_2, c1_2, c5_3, c5_1, c5_2, c6_1 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 8 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0)))))) S tuples: 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0)))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1, c5 ---------------------------------------- (55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 1'(0(0(1(0(1(x0)))))) -> c5 We considered the (Usable) Rules:none And the Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0)))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0(x_1)) = 0 POL(1(x_1)) = 0 POL(1'(x_1)) = [1] POL(ENCARG(x_1)) = x_1 POL(ENCODE_1(x_1)) = x_1 POL(c(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c5) = 0 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(cons_0(x_1)) = [1] + x_1 POL(cons_1(x_1)) = [1] + x_1 POL(encArg(x_1)) = [1] ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0)))))) S tuples: 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0)))))) K tuples: 1'(0(0(1(0(1(x0)))))) -> c5 Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1, c5 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(0(z0)))))) by 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(1(z0)))))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) 1'(0(0(1(0(1(x0)))))) -> c5 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(1(z0)))))) S tuples: 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(1(z0)))))) K tuples: 1'(0(0(1(0(1(x0)))))) -> c5 Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1, c5 ---------------------------------------- (59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: 1'(0(0(1(0(1(x0)))))) -> c5 ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(1(z0)))))) S tuples: 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5(1'(0(0(0(1(z0)))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5 S tuples: 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5 K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1, c5 ---------------------------------------- (63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 1'(0(0(1(0(0(0(z0))))))) -> c5 We considered the (Usable) Rules:none And the Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5 The order we found is given by the following interpretation: Polynomial interpretation : POL(0(x_1)) = [1] POL(1(x_1)) = 0 POL(1'(x_1)) = [1] POL(ENCARG(x_1)) = x_1 POL(ENCODE_1(x_1)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c5) = 0 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(cons_0(x_1)) = [1] + x_1 POL(cons_1(x_1)) = [1] + x_1 POL(encArg(x_1)) = [1] ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5 S tuples: 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) K tuples: 1'(0(0(1(0(0(0(z0))))))) -> c5 Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1, c5 ---------------------------------------- (65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_0(cons_0(cons_0(z0)))) -> c(ENCARG(cons_0(cons_0(z0)))) by ENCARG(cons_0(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_0(cons_0(cons_1(y0))))) -> c(ENCARG(cons_0(cons_0(cons_1(y0))))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) 1'(0(0(1(0(0(0(z0))))))) -> c5 ENCARG(cons_0(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_0(cons_0(cons_1(y0))))) -> c(ENCARG(cons_0(cons_0(cons_1(y0))))) S tuples: 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) K tuples: 1'(0(0(1(0(0(0(z0))))))) -> c5 Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1, c5 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: 1'(0(0(1(0(0(0(z0))))))) -> c5 ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) ENCARG(cons_0(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_0(cons_0(cons_1(y0))))) -> c(ENCARG(cons_0(cons_0(cons_1(y0))))) S tuples: 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1 ---------------------------------------- (69) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_0(cons_0(cons_1(z0)))) -> c(ENCARG(cons_0(cons_1(z0)))) by ENCARG(cons_0(cons_0(cons_1(cons_0(y0))))) -> c(ENCARG(cons_0(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_0(cons_1(cons_1(y0))))) -> c(ENCARG(cons_0(cons_1(cons_1(y0))))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) ENCARG(cons_0(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_0(cons_0(cons_1(y0))))) -> c(ENCARG(cons_0(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_0(cons_1(cons_0(y0))))) -> c(ENCARG(cons_0(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_0(cons_1(cons_1(y0))))) -> c(ENCARG(cons_0(cons_1(cons_1(y0))))) S tuples: 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1 ---------------------------------------- (71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_0(cons_1(cons_0(z0)))) -> c(ENCARG(cons_1(cons_0(z0)))) by ENCARG(cons_0(cons_1(cons_0(cons_0(y0))))) -> c(ENCARG(cons_1(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_1(y0))))) -> c(ENCARG(cons_1(cons_0(cons_1(y0))))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0)))) 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) ENCARG(cons_0(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_0(cons_0(cons_1(y0))))) -> c(ENCARG(cons_0(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_0(cons_1(cons_0(y0))))) -> c(ENCARG(cons_0(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_0(cons_1(cons_1(y0))))) -> c(ENCARG(cons_0(cons_1(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_0(y0))))) -> c(ENCARG(cons_1(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_1(y0))))) -> c(ENCARG(cons_1(cons_0(cons_1(y0))))) S tuples: 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1 ---------------------------------------- (73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_0(cons_1(cons_1(z0)))) -> c(ENCARG(cons_1(cons_1(z0)))) by ENCARG(cons_0(cons_1(cons_1(cons_0(y0))))) -> c(ENCARG(cons_1(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_1(y0))))) -> c(ENCARG(cons_1(cons_1(cons_1(y0))))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) ENCARG(cons_0(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_0(cons_0(cons_1(y0))))) -> c(ENCARG(cons_0(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_0(cons_1(cons_0(y0))))) -> c(ENCARG(cons_0(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_0(cons_1(cons_1(y0))))) -> c(ENCARG(cons_0(cons_1(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_0(y0))))) -> c(ENCARG(cons_1(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_1(y0))))) -> c(ENCARG(cons_1(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_0(y0))))) -> c(ENCARG(cons_1(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_1(y0))))) -> c(ENCARG(cons_1(cons_1(cons_1(y0))))) S tuples: 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c5_1, c_1 ---------------------------------------- (75) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace 1'(0(0(1(0(1(z0)))))) -> c5(1'(0(0(1(z0))))) by 1'(0(0(1(0(1(0(1(y0)))))))) -> c5(1'(0(0(1(0(1(y0))))))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_0(cons_0(cons_1(y0))))) -> c(ENCARG(cons_0(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_0(cons_1(cons_0(y0))))) -> c(ENCARG(cons_0(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_0(cons_1(cons_1(y0))))) -> c(ENCARG(cons_0(cons_1(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_0(y0))))) -> c(ENCARG(cons_1(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_1(y0))))) -> c(ENCARG(cons_1(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_0(y0))))) -> c(ENCARG(cons_1(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_1(y0))))) -> c(ENCARG(cons_1(cons_1(cons_1(y0))))) 1'(0(0(1(0(1(0(1(y0)))))))) -> c5(1'(0(0(1(0(1(y0))))))) S tuples: 1'(0(0(1(0(1(0(1(y0)))))))) -> c5(1'(0(0(1(0(1(y0))))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1 ---------------------------------------- (77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_0(cons_0(cons_0(cons_0(y0))))) -> c(ENCARG(cons_0(cons_0(cons_0(y0))))) by ENCARG(cons_0(cons_0(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_1(y0)))))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_0(cons_0(cons_1(y0))))) -> c(ENCARG(cons_0(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_0(cons_1(cons_0(y0))))) -> c(ENCARG(cons_0(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_0(cons_1(cons_1(y0))))) -> c(ENCARG(cons_0(cons_1(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_0(y0))))) -> c(ENCARG(cons_1(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_1(y0))))) -> c(ENCARG(cons_1(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_0(y0))))) -> c(ENCARG(cons_1(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_1(y0))))) -> c(ENCARG(cons_1(cons_1(cons_1(y0))))) 1'(0(0(1(0(1(0(1(y0)))))))) -> c5(1'(0(0(1(0(1(y0))))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_1(y0)))))) S tuples: 1'(0(0(1(0(1(0(1(y0)))))))) -> c5(1'(0(0(1(0(1(y0))))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1 ---------------------------------------- (79) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_0(cons_0(cons_0(cons_1(y0))))) -> c(ENCARG(cons_0(cons_0(cons_1(y0))))) by ENCARG(cons_0(cons_0(cons_0(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_1(y0)))))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_0(cons_1(cons_0(y0))))) -> c(ENCARG(cons_0(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_0(cons_1(cons_1(y0))))) -> c(ENCARG(cons_0(cons_1(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_0(y0))))) -> c(ENCARG(cons_1(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_1(y0))))) -> c(ENCARG(cons_1(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_0(y0))))) -> c(ENCARG(cons_1(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_1(y0))))) -> c(ENCARG(cons_1(cons_1(cons_1(y0))))) 1'(0(0(1(0(1(0(1(y0)))))))) -> c5(1'(0(0(1(0(1(y0))))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_1(y0)))))) S tuples: 1'(0(0(1(0(1(0(1(y0)))))))) -> c5(1'(0(0(1(0(1(y0))))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1 ---------------------------------------- (81) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_0(cons_0(cons_1(cons_0(y0))))) -> c(ENCARG(cons_0(cons_1(cons_0(y0))))) by ENCARG(cons_0(cons_0(cons_1(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_1(y0)))))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_0(cons_1(cons_1(y0))))) -> c(ENCARG(cons_0(cons_1(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_0(y0))))) -> c(ENCARG(cons_1(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_1(y0))))) -> c(ENCARG(cons_1(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_0(y0))))) -> c(ENCARG(cons_1(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_1(y0))))) -> c(ENCARG(cons_1(cons_1(cons_1(y0))))) 1'(0(0(1(0(1(0(1(y0)))))))) -> c5(1'(0(0(1(0(1(y0))))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_1(y0)))))) S tuples: 1'(0(0(1(0(1(0(1(y0)))))))) -> c5(1'(0(0(1(0(1(y0))))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1 ---------------------------------------- (83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_0(cons_0(cons_1(cons_1(y0))))) -> c(ENCARG(cons_0(cons_1(cons_1(y0))))) by ENCARG(cons_0(cons_0(cons_1(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_1(cons_1(y0)))))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_1(cons_0(cons_0(y0))))) -> c(ENCARG(cons_1(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_1(y0))))) -> c(ENCARG(cons_1(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_0(y0))))) -> c(ENCARG(cons_1(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_1(y0))))) -> c(ENCARG(cons_1(cons_1(cons_1(y0))))) 1'(0(0(1(0(1(0(1(y0)))))))) -> c5(1'(0(0(1(0(1(y0))))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_1(cons_1(y0)))))) S tuples: 1'(0(0(1(0(1(0(1(y0)))))))) -> c5(1'(0(0(1(0(1(y0))))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1 ---------------------------------------- (85) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace 1'(0(0(1(0(1(0(1(y0)))))))) -> c5(1'(0(0(1(0(1(y0))))))) by 1'(0(0(1(0(1(0(1(0(1(y0)))))))))) -> c5(1'(0(0(1(0(1(0(1(y0))))))))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_1(cons_0(cons_0(y0))))) -> c(ENCARG(cons_1(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_1(y0))))) -> c(ENCARG(cons_1(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_0(y0))))) -> c(ENCARG(cons_1(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_1(y0))))) -> c(ENCARG(cons_1(cons_1(cons_1(y0))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_1(cons_1(y0)))))) 1'(0(0(1(0(1(0(1(0(1(y0)))))))))) -> c5(1'(0(0(1(0(1(0(1(y0))))))))) S tuples: 1'(0(0(1(0(1(0(1(0(1(y0)))))))))) -> c5(1'(0(0(1(0(1(0(1(y0))))))))) K tuples:none Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1 ---------------------------------------- (87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCARG(cons_0(cons_1(cons_0(cons_0(y0))))) -> c(ENCARG(cons_1(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_1(y0))))) -> c(ENCARG(cons_1(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_0(y0))))) -> c(ENCARG(cons_1(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_1(y0))))) -> c(ENCARG(cons_1(cons_1(cons_1(y0))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_1(cons_1(y0)))))) 1'(0(0(1(0(1(0(1(0(1(y0)))))))))) -> c5(1'(0(0(1(0(1(0(1(y0))))))))) We considered the (Usable) Rules: 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) encArg(cons_1(z0)) -> 1(encArg(z0)) encArg(cons_0(z0)) -> 0(encArg(z0)) And the Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_1(cons_0(cons_0(y0))))) -> c(ENCARG(cons_1(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_1(y0))))) -> c(ENCARG(cons_1(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_0(y0))))) -> c(ENCARG(cons_1(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_1(y0))))) -> c(ENCARG(cons_1(cons_1(cons_1(y0))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_1(cons_1(y0)))))) 1'(0(0(1(0(1(0(1(0(1(y0)))))))))) -> c5(1'(0(0(1(0(1(0(1(y0))))))))) The order we found is given by the following interpretation: Matrix interpretation [MATRO]: Non-tuple symbols: <<< M( 0_1(x_1) ) = [[0], [2]] + [[0, 0], [0, 1]] * x_1 >>> <<< M( cons_1_1(x_1) ) = [[2], [2]] + [[1, 4], [0, 1]] * x_1 >>> <<< M( 1_1(x_1) ) = [[1], [2]] + [[0, 0], [0, 1]] * x_1 >>> <<< M( cons_0_1(x_1) ) = [[4], [4]] + [[1, 0], [0, 1]] * x_1 >>> <<< M( encArg_1(x_1) ) = [[0], [0]] + [[0, 4], [0, 1]] * x_1 >>> Tuple symbols: <<< M( 1'_1(x_1) ) = [[0], [4]] + [[0, 4], [0, 0]] * x_1 >>> <<< M( ENCODE_1_1(x_1) ) = [[0], [4]] + [[0, 4], [0, 1]] * x_1 >>> <<< M( c_1(x_1) ) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< M( c1_2(x_1, x_2) ) = [[2], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 >>> <<< M( ENCARG_1(x_1) ) = [[4], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< M( c5_1(x_1) ) = [[0], [0]] + [[1, 2], [0, 0]] * x_1 >>> <<< M( c6_1(x_1) ) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> Matrix type: We used a basic matrix type which is not further parametrizeable. As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order. ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_0(z0)) -> 0(encArg(z0)) encArg(cons_1(z0)) -> 1(encArg(z0)) 0(0(0(0(z0)))) -> 0(0(0(1(z0)))) 1(0(0(1(z0)))) -> 0(0(1(0(z0)))) Tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCARG(cons_1(cons_1(cons_1(z0)))) -> c1(1'(1(1(encArg(z0)))), ENCARG(cons_1(cons_1(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCODE_1(cons_1(cons_1(z0))) -> c6(1'(1(1(encArg(z0))))) ENCARG(cons_0(cons_1(cons_0(cons_0(y0))))) -> c(ENCARG(cons_1(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_1(y0))))) -> c(ENCARG(cons_1(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_0(y0))))) -> c(ENCARG(cons_1(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_1(y0))))) -> c(ENCARG(cons_1(cons_1(cons_1(y0))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_1(cons_1(y0)))))) 1'(0(0(1(0(1(0(1(0(1(y0)))))))))) -> c5(1'(0(0(1(0(1(0(1(y0))))))))) S tuples:none K tuples: ENCARG(cons_1(cons_0(cons_0(z0)))) -> c1(1'(0(0(encArg(z0)))), ENCARG(cons_0(cons_0(z0)))) ENCARG(cons_1(cons_0(cons_1(z0)))) -> c1(1'(0(1(encArg(z0)))), ENCARG(cons_0(cons_1(z0)))) ENCARG(cons_1(cons_1(cons_0(z0)))) -> c1(1'(1(0(encArg(z0)))), ENCARG(cons_1(cons_0(z0)))) ENCODE_1(cons_0(cons_0(z0))) -> c6(1'(0(0(encArg(z0))))) ENCODE_1(cons_0(cons_1(z0))) -> c6(1'(0(1(encArg(z0))))) ENCODE_1(cons_1(cons_0(z0))) -> c6(1'(1(0(encArg(z0))))) ENCARG(cons_0(cons_1(cons_0(cons_0(y0))))) -> c(ENCARG(cons_1(cons_0(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_0(cons_1(y0))))) -> c(ENCARG(cons_1(cons_0(cons_1(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_0(y0))))) -> c(ENCARG(cons_1(cons_1(cons_0(y0))))) ENCARG(cons_0(cons_1(cons_1(cons_1(y0))))) -> c(ENCARG(cons_1(cons_1(cons_1(y0))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_0(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_0(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_0(cons_1(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_0(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_0(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_0(cons_1(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_1(cons_0(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_1(cons_0(y0)))))) ENCARG(cons_0(cons_0(cons_1(cons_1(cons_1(y0)))))) -> c(ENCARG(cons_0(cons_1(cons_1(cons_1(y0)))))) 1'(0(0(1(0(1(0(1(0(1(y0)))))))))) -> c5(1'(0(0(1(0(1(0(1(y0))))))))) Defined Rule Symbols: encArg_1, 0_1, 1_1 Defined Pair Symbols: ENCARG_1, ENCODE_1_1, 1'_1 Compound Symbols: c1_2, c6_1, c_1, c5_1 ---------------------------------------- (89) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (90) BOUNDS(1, 1)