/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 172 ms] (4) CpxRelTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (6) CdtProblem (7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 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) CdtRhsSimplificationProcessorProof [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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 26 ms] (72) CdtProblem (73) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (74) BOUNDS(1, 1) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: a(a(x1)) -> b(x1) b(a(c(x1))) -> c(c(a(a(a(a(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(c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: a(a(x1)) -> b(x1) b(a(c(x1))) -> c(c(a(a(a(a(x1)))))) The (relative) TRS S consists of the following rules: encArg(c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(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^1). The TRS R consists of the following rules: a(a(x1)) -> b(x1) b(a(c(x1))) -> c(c(a(a(a(a(x1)))))) The (relative) TRS S consists of the following rules: encArg(c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(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(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_c(z0) -> c(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0)) ENCODE_A(z0) -> c4(A(encArg(z0)), ENCARG(z0)) ENCODE_B(z0) -> c5(B(encArg(z0)), ENCARG(z0)) ENCODE_C(z0) -> c6(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: a_1, b_1, encArg_1, encode_a_1, encode_b_1, encode_c_1 Defined Pair Symbols: ENCARG_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, A_1, B_1 Compound Symbols: c1_1, c2_2, c3_2, c4_2, c5_2, c6_1, c7_1, c8_4 ---------------------------------------- (7) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ENCODE_C(z0) -> c6(ENCARG(z0)) ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_c(z0) -> c(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0)) ENCODE_A(z0) -> c4(A(encArg(z0)), ENCARG(z0)) ENCODE_B(z0) -> c5(B(encArg(z0)), ENCARG(z0)) A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: a_1, b_1, encArg_1, encode_a_1, encode_b_1, encode_c_1 Defined Pair Symbols: ENCARG_1, ENCODE_A_1, ENCODE_B_1, A_1, B_1 Compound Symbols: c1_1, c2_2, c3_2, c4_2, c5_2, c7_1, c8_4 ---------------------------------------- (9) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_c(z0) -> c(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0)) A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) ENCODE_A(z0) -> c6(A(encArg(z0))) ENCODE_A(z0) -> c6(ENCARG(z0)) ENCODE_B(z0) -> c6(B(encArg(z0))) ENCODE_B(z0) -> c6(ENCARG(z0)) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: a_1, b_1, encArg_1, encode_a_1, encode_b_1, encode_c_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c2_2, c3_2, c7_1, c8_4, c6_1 ---------------------------------------- (11) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ENCODE_A(z0) -> c6(ENCARG(z0)) ENCODE_B(z0) -> c6(ENCARG(z0)) ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) encode_a(z0) -> a(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_c(z0) -> c(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0)) A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) ENCODE_A(z0) -> c6(A(encArg(z0))) ENCODE_B(z0) -> c6(B(encArg(z0))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: a_1, b_1, encArg_1, encode_a_1, encode_b_1, encode_c_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c2_2, c3_2, c7_1, c8_4, c6_1 ---------------------------------------- (13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_a(z0) -> a(encArg(z0)) encode_b(z0) -> b(encArg(z0)) encode_c(z0) -> c(encArg(z0)) ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0)) A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) ENCODE_A(z0) -> c6(A(encArg(z0))) ENCODE_B(z0) -> c6(B(encArg(z0))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c2_2, c3_2, c7_1, c8_4, c6_1 ---------------------------------------- (15) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0)) by ENCARG(cons_a(c(z0))) -> c2(A(c(encArg(z0))), ENCARG(c(z0))) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c2(A(b(encArg(z0))), ENCARG(cons_b(z0))) ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0)) A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) ENCODE_A(z0) -> c6(A(encArg(z0))) ENCODE_B(z0) -> c6(B(encArg(z0))) ENCARG(cons_a(c(z0))) -> c2(A(c(encArg(z0))), ENCARG(c(z0))) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c2(A(b(encArg(z0))), ENCARG(cons_b(z0))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c3_2, c7_1, c8_4, c6_1, c2_2 ---------------------------------------- (17) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0)) A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) ENCODE_A(z0) -> c6(A(encArg(z0))) ENCODE_B(z0) -> c6(B(encArg(z0))) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c2(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c3_2, c7_1, c8_4, c6_1, c2_2, c2_1 ---------------------------------------- (19) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(z0)) -> c3(B(encArg(z0)), ENCARG(z0)) by ENCARG(cons_b(c(z0))) -> c3(B(c(encArg(z0))), ENCARG(c(z0))) ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0))) ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) ENCODE_A(z0) -> c6(A(encArg(z0))) ENCODE_B(z0) -> c6(B(encArg(z0))) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c2(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(B(c(encArg(z0))), ENCARG(c(z0))) ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c7_1, c8_4, c6_1, c2_2, c2_1, c3_2 ---------------------------------------- (21) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) ENCODE_A(z0) -> c6(A(encArg(z0))) ENCODE_B(z0) -> c6(B(encArg(z0))) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c2(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c7_1, c8_4, c6_1, c2_2, c2_1, c3_2, c3_1 ---------------------------------------- (23) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(a(c(z0))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0)), A(z0)) by B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCODE_A(z0) -> c6(A(encArg(z0))) ENCODE_B(z0) -> c6(B(encArg(z0))) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c2(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, ENCODE_A_1, ENCODE_B_1, B_1 Compound Symbols: c1_1, c7_1, c6_1, c2_2, c2_1, c3_2, c3_1, c8_4 ---------------------------------------- (25) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_A(z0) -> c6(A(encArg(z0))) by ENCODE_A(c(z0)) -> c6(A(c(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCODE_B(z0) -> c6(B(encArg(z0))) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c2(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) ENCODE_A(c(z0)) -> c6(A(c(encArg(z0)))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, ENCODE_B_1, B_1, ENCODE_A_1 Compound Symbols: c1_1, c7_1, c6_1, c2_2, c2_1, c3_2, c3_1, c8_4 ---------------------------------------- (27) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_A(c(z0)) -> c6(A(c(encArg(z0)))) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCODE_B(z0) -> c6(B(encArg(z0))) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c2(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, ENCODE_B_1, B_1, ENCODE_A_1 Compound Symbols: c1_1, c7_1, c6_1, c2_2, c2_1, c3_2, c3_1, c8_4 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_B(z0) -> c6(B(encArg(z0))) by ENCODE_B(c(z0)) -> c6(B(c(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c2(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_B(c(z0)) -> c6(B(c(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_2, c3_1, c8_4, c6_1 ---------------------------------------- (31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_B(c(z0)) -> c6(B(c(encArg(z0)))) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(z0))) -> c2(A(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_2, c3_1, c8_4, c6_1 ---------------------------------------- (33) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_a(cons_b(z0))) -> c2(A(b(encArg(z0))), ENCARG(cons_b(z0))) by ENCARG(cons_a(cons_b(c(z0)))) -> c2(A(b(c(encArg(z0)))), ENCARG(cons_b(c(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(A(b(c(encArg(z0)))), ENCARG(cons_b(c(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_2, c3_1, c8_4, c6_1 ---------------------------------------- (35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_2, c3_1, c8_4, c6_1 ---------------------------------------- (37) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(cons_a(z0))) -> c3(B(a(encArg(z0))), ENCARG(cons_a(z0))) by ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_2, c3_1, c8_4, c6_1 ---------------------------------------- (39) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_b(cons_b(z0))) -> c3(B(b(encArg(z0))), ENCARG(cons_b(z0))) by ENCARG(cons_b(cons_b(c(z0)))) -> c3(B(b(c(encArg(z0)))), ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(B(b(c(encArg(z0)))), ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_1, c8_4, c6_1, c3_2 ---------------------------------------- (41) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_1, c8_4, c6_1, c3_2 ---------------------------------------- (43) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(a(c(x0))) -> c8(A(b(a(x0))), A(a(a(x0))), A(a(x0)), A(x0)) by B(a(c(c(z0)))) -> c8(A(c(c(a(a(a(a(z0))))))), A(a(a(c(z0)))), A(a(c(z0))), A(c(z0))) B(a(c(a(z0)))) -> c8(A(b(b(z0))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) B(a(c(c(z0)))) -> c8(A(c(c(a(a(a(a(z0))))))), A(a(a(c(z0)))), A(a(c(z0))), A(c(z0))) B(a(c(a(z0)))) -> c8(A(b(b(z0))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(c(z0)))) -> c8(A(c(c(a(a(a(a(z0))))))), A(a(a(c(z0)))), A(a(c(z0))), A(c(z0))) B(a(c(a(z0)))) -> c8(A(b(b(z0))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_1, c8_4, c6_1, c3_2, c8_2 ---------------------------------------- (45) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_1, c8_4, c6_1, c3_2, c8_2, c8_3 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(a(c(z0))) -> c8(A(a(b(z0))), A(a(a(z0))), A(a(z0)), A(z0)) by B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, B_1, ENCODE_A_1, ENCODE_B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_1, c8_4, c6_1, c3_2, c8_2, c8_3, c8_1 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(a(c(a(z0)))) -> c8(A(a(a(b(z0)))), A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) by B(a(c(a(x0)))) -> c8(A(b(b(x0))), A(a(a(a(x0)))), A(a(a(x0))), A(a(x0))) B(a(c(a(x0)))) -> c8(A(a(a(a(x0)))), A(a(a(x0))), A(a(x0))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) B(a(c(a(x0)))) -> c8(A(b(b(x0))), A(a(a(a(x0)))), A(a(a(x0))), A(a(x0))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) B(a(c(a(x0)))) -> c8(A(b(b(x0))), A(a(a(a(x0)))), A(a(a(x0))), A(a(x0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, ENCODE_A_1, ENCODE_B_1, B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_1, c6_1, c3_2, c8_2, c8_3, c8_4, c8_1 ---------------------------------------- (51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, ENCODE_A_1, ENCODE_B_1, B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_1, c6_1, c3_2, c8_2, c8_3, c8_4, c8_1 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_A(cons_b(z0)) -> c6(A(b(encArg(z0)))) by ENCODE_A(cons_b(c(z0))) -> c6(A(b(c(encArg(z0))))) ENCODE_A(cons_b(cons_a(z0))) -> c6(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c6(A(b(b(encArg(z0))))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) ENCODE_A(cons_b(c(z0))) -> c6(A(b(c(encArg(z0))))) ENCODE_A(cons_b(cons_a(z0))) -> c6(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c6(A(b(b(encArg(z0))))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, ENCODE_A_1, ENCODE_B_1, B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_1, c6_1, c3_2, c8_2, c8_3, c8_4, c8_1 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_A(cons_b(c(z0))) -> c6(A(b(c(encArg(z0))))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) ENCODE_A(cons_b(cons_a(z0))) -> c6(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c6(A(b(b(encArg(z0))))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, ENCODE_A_1, ENCODE_B_1, B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_1, c6_1, c3_2, c8_2, c8_3, c8_4, c8_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_B(cons_a(z0)) -> c6(B(a(encArg(z0)))) by ENCODE_B(cons_a(c(z0))) -> c6(B(a(c(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c6(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c6(B(a(b(encArg(z0))))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) ENCODE_A(cons_b(cons_a(z0))) -> c6(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c6(A(b(b(encArg(z0))))) ENCODE_B(cons_a(c(z0))) -> c6(B(a(c(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c6(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c6(B(a(b(encArg(z0))))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, ENCODE_A_1, ENCODE_B_1, B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_1, c6_1, c3_2, c8_2, c8_3, c8_4, c8_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_B(cons_b(z0)) -> c6(B(b(encArg(z0)))) by ENCODE_B(cons_b(c(z0))) -> c6(B(b(c(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c6(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c6(B(b(b(encArg(z0))))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) ENCODE_A(cons_b(cons_a(z0))) -> c6(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c6(A(b(b(encArg(z0))))) ENCODE_B(cons_a(c(z0))) -> c6(B(a(c(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c6(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c6(B(a(b(encArg(z0))))) ENCODE_B(cons_b(c(z0))) -> c6(B(b(c(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c6(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c6(B(b(b(encArg(z0))))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, ENCODE_A_1, B_1, ENCODE_B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_1, c6_1, c3_2, c8_2, c8_3, c8_4, c8_1 ---------------------------------------- (61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_B(cons_b(c(z0))) -> c6(B(b(c(encArg(z0))))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) ENCODE_A(cons_b(cons_a(z0))) -> c6(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c6(A(b(b(encArg(z0))))) ENCODE_B(cons_a(c(z0))) -> c6(B(a(c(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c6(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c6(B(a(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c6(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c6(B(b(b(encArg(z0))))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, A_1, ENCODE_A_1, B_1, ENCODE_B_1 Compound Symbols: c1_1, c7_1, c2_2, c2_1, c3_1, c6_1, c3_2, c8_2, c8_3, c8_4, c8_1 ---------------------------------------- (63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(c(z0)) -> c1(ENCARG(z0)) by ENCARG(c(c(y0))) -> c1(ENCARG(c(y0))) ENCARG(c(cons_a(cons_a(y0)))) -> c1(ENCARG(cons_a(cons_a(y0)))) ENCARG(c(cons_a(c(y0)))) -> c1(ENCARG(cons_a(c(y0)))) ENCARG(c(cons_b(c(y0)))) -> c1(ENCARG(cons_b(c(y0)))) ENCARG(c(cons_a(cons_b(cons_a(y0))))) -> c1(ENCARG(cons_a(cons_b(cons_a(y0))))) ENCARG(c(cons_a(cons_b(cons_b(y0))))) -> c1(ENCARG(cons_a(cons_b(cons_b(y0))))) ENCARG(c(cons_a(cons_b(c(y0))))) -> c1(ENCARG(cons_a(cons_b(c(y0))))) ENCARG(c(cons_b(cons_a(c(y0))))) -> c1(ENCARG(cons_b(cons_a(c(y0))))) ENCARG(c(cons_b(cons_a(cons_a(y0))))) -> c1(ENCARG(cons_b(cons_a(cons_a(y0))))) ENCARG(c(cons_b(cons_a(cons_b(y0))))) -> c1(ENCARG(cons_b(cons_a(cons_b(y0))))) ENCARG(c(cons_b(cons_b(cons_a(y0))))) -> c1(ENCARG(cons_b(cons_b(cons_a(y0))))) ENCARG(c(cons_b(cons_b(cons_b(y0))))) -> c1(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(c(cons_b(cons_b(c(y0))))) -> c1(ENCARG(cons_b(cons_b(c(y0))))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: A(a(z0)) -> c7(B(z0)) ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) ENCODE_A(cons_b(cons_a(z0))) -> c6(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c6(A(b(b(encArg(z0))))) ENCODE_B(cons_a(c(z0))) -> c6(B(a(c(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c6(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c6(B(a(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c6(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c6(B(b(b(encArg(z0))))) ENCARG(c(c(y0))) -> c1(ENCARG(c(y0))) ENCARG(c(cons_a(cons_a(y0)))) -> c1(ENCARG(cons_a(cons_a(y0)))) ENCARG(c(cons_a(c(y0)))) -> c1(ENCARG(cons_a(c(y0)))) ENCARG(c(cons_b(c(y0)))) -> c1(ENCARG(cons_b(c(y0)))) ENCARG(c(cons_a(cons_b(cons_a(y0))))) -> c1(ENCARG(cons_a(cons_b(cons_a(y0))))) ENCARG(c(cons_a(cons_b(cons_b(y0))))) -> c1(ENCARG(cons_a(cons_b(cons_b(y0))))) ENCARG(c(cons_a(cons_b(c(y0))))) -> c1(ENCARG(cons_a(cons_b(c(y0))))) ENCARG(c(cons_b(cons_a(c(y0))))) -> c1(ENCARG(cons_b(cons_a(c(y0))))) ENCARG(c(cons_b(cons_a(cons_a(y0))))) -> c1(ENCARG(cons_b(cons_a(cons_a(y0))))) ENCARG(c(cons_b(cons_a(cons_b(y0))))) -> c1(ENCARG(cons_b(cons_a(cons_b(y0))))) ENCARG(c(cons_b(cons_b(cons_a(y0))))) -> c1(ENCARG(cons_b(cons_b(cons_a(y0))))) ENCARG(c(cons_b(cons_b(cons_b(y0))))) -> c1(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(c(cons_b(cons_b(c(y0))))) -> c1(ENCARG(cons_b(cons_b(c(y0))))) S tuples: A(a(z0)) -> c7(B(z0)) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: A_1, ENCARG_1, ENCODE_A_1, B_1, ENCODE_B_1 Compound Symbols: c7_1, c2_2, c2_1, c3_1, c6_1, c3_2, c8_2, c8_3, c8_4, c8_1, c1_1 ---------------------------------------- (65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A(a(z0)) -> c7(B(z0)) by A(a(a(c(y0)))) -> c7(B(a(c(y0)))) A(a(a(c(c(y0))))) -> c7(B(a(c(c(y0))))) A(a(a(c(a(y0))))) -> c7(B(a(c(a(y0))))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) ENCODE_A(cons_b(cons_a(z0))) -> c6(A(b(a(encArg(z0))))) ENCODE_A(cons_b(cons_b(z0))) -> c6(A(b(b(encArg(z0))))) ENCODE_B(cons_a(c(z0))) -> c6(B(a(c(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c6(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c6(B(a(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c6(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c6(B(b(b(encArg(z0))))) ENCARG(c(c(y0))) -> c1(ENCARG(c(y0))) ENCARG(c(cons_a(cons_a(y0)))) -> c1(ENCARG(cons_a(cons_a(y0)))) ENCARG(c(cons_a(c(y0)))) -> c1(ENCARG(cons_a(c(y0)))) ENCARG(c(cons_b(c(y0)))) -> c1(ENCARG(cons_b(c(y0)))) ENCARG(c(cons_a(cons_b(cons_a(y0))))) -> c1(ENCARG(cons_a(cons_b(cons_a(y0))))) ENCARG(c(cons_a(cons_b(cons_b(y0))))) -> c1(ENCARG(cons_a(cons_b(cons_b(y0))))) ENCARG(c(cons_a(cons_b(c(y0))))) -> c1(ENCARG(cons_a(cons_b(c(y0))))) ENCARG(c(cons_b(cons_a(c(y0))))) -> c1(ENCARG(cons_b(cons_a(c(y0))))) ENCARG(c(cons_b(cons_a(cons_a(y0))))) -> c1(ENCARG(cons_b(cons_a(cons_a(y0))))) ENCARG(c(cons_b(cons_a(cons_b(y0))))) -> c1(ENCARG(cons_b(cons_a(cons_b(y0))))) ENCARG(c(cons_b(cons_b(cons_a(y0))))) -> c1(ENCARG(cons_b(cons_b(cons_a(y0))))) ENCARG(c(cons_b(cons_b(cons_b(y0))))) -> c1(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(c(cons_b(cons_b(c(y0))))) -> c1(ENCARG(cons_b(cons_b(c(y0))))) A(a(a(c(y0)))) -> c7(B(a(c(y0)))) A(a(a(c(c(y0))))) -> c7(B(a(c(c(y0))))) A(a(a(c(a(y0))))) -> c7(B(a(c(a(y0))))) S tuples: B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) A(a(a(c(y0)))) -> c7(B(a(c(y0)))) A(a(a(c(c(y0))))) -> c7(B(a(c(c(y0))))) A(a(a(c(a(y0))))) -> c7(B(a(c(a(y0))))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, ENCODE_A_1, B_1, ENCODE_B_1, A_1 Compound Symbols: c2_2, c2_1, c3_1, c6_1, c3_2, c8_2, c8_3, c8_4, c8_1, c1_1, c7_1 ---------------------------------------- (67) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 3 leading nodes: A(a(a(c(y0)))) -> c7(B(a(c(y0)))) A(a(a(c(c(y0))))) -> c7(B(a(c(c(y0))))) A(a(a(c(a(y0))))) -> c7(B(a(c(a(y0))))) Removed 3 trailing nodes: ENCODE_A(cons_b(cons_a(z0))) -> c6(A(b(a(encArg(z0))))) ENCODE_A(cons_a(z0)) -> c6(A(a(encArg(z0)))) ENCODE_A(cons_b(cons_b(z0))) -> c6(A(b(b(encArg(z0))))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(cons_a(cons_a(z0))) -> c2(A(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(A(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(A(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) ENCODE_B(cons_a(c(z0))) -> c6(B(a(c(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c6(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c6(B(a(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c6(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c6(B(b(b(encArg(z0))))) ENCARG(c(c(y0))) -> c1(ENCARG(c(y0))) ENCARG(c(cons_a(cons_a(y0)))) -> c1(ENCARG(cons_a(cons_a(y0)))) ENCARG(c(cons_a(c(y0)))) -> c1(ENCARG(cons_a(c(y0)))) ENCARG(c(cons_b(c(y0)))) -> c1(ENCARG(cons_b(c(y0)))) ENCARG(c(cons_a(cons_b(cons_a(y0))))) -> c1(ENCARG(cons_a(cons_b(cons_a(y0))))) ENCARG(c(cons_a(cons_b(cons_b(y0))))) -> c1(ENCARG(cons_a(cons_b(cons_b(y0))))) ENCARG(c(cons_a(cons_b(c(y0))))) -> c1(ENCARG(cons_a(cons_b(c(y0))))) ENCARG(c(cons_b(cons_a(c(y0))))) -> c1(ENCARG(cons_b(cons_a(c(y0))))) ENCARG(c(cons_b(cons_a(cons_a(y0))))) -> c1(ENCARG(cons_b(cons_a(cons_a(y0))))) ENCARG(c(cons_b(cons_a(cons_b(y0))))) -> c1(ENCARG(cons_b(cons_a(cons_b(y0))))) ENCARG(c(cons_b(cons_b(cons_a(y0))))) -> c1(ENCARG(cons_b(cons_b(cons_a(y0))))) ENCARG(c(cons_b(cons_b(cons_b(y0))))) -> c1(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(c(cons_b(cons_b(c(y0))))) -> c1(ENCARG(cons_b(cons_b(c(y0))))) S tuples: B(a(c(x0))) -> c8(A(a(a(x0))), A(a(x0))) B(a(c(c(z0)))) -> c8(A(a(a(c(z0)))), A(a(c(z0)))) B(a(c(a(z0)))) -> c8(A(a(a(a(z0)))), A(a(a(z0))), A(a(z0))) B(a(c(z0))) -> c8(A(a(b(z0))), A(b(z0)), A(a(z0)), A(z0)) B(a(c(a(z0)))) -> c8(A(a(b(a(z0)))), A(a(b(z0))), A(a(a(z0))), A(a(z0))) B(a(c(x0))) -> c8(A(a(b(x0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, B_1, ENCODE_B_1 Compound Symbols: c2_2, c2_1, c3_1, c3_2, c8_2, c8_3, c8_4, c8_1, c6_1, c1_1 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 25 trailing tuple parts ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) ENCODE_B(cons_a(c(z0))) -> c6(B(a(c(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c6(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c6(B(a(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c6(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c6(B(b(b(encArg(z0))))) ENCARG(c(c(y0))) -> c1(ENCARG(c(y0))) ENCARG(c(cons_a(cons_a(y0)))) -> c1(ENCARG(cons_a(cons_a(y0)))) ENCARG(c(cons_a(c(y0)))) -> c1(ENCARG(cons_a(c(y0)))) ENCARG(c(cons_b(c(y0)))) -> c1(ENCARG(cons_b(c(y0)))) ENCARG(c(cons_a(cons_b(cons_a(y0))))) -> c1(ENCARG(cons_a(cons_b(cons_a(y0))))) ENCARG(c(cons_a(cons_b(cons_b(y0))))) -> c1(ENCARG(cons_a(cons_b(cons_b(y0))))) ENCARG(c(cons_a(cons_b(c(y0))))) -> c1(ENCARG(cons_a(cons_b(c(y0))))) ENCARG(c(cons_b(cons_a(c(y0))))) -> c1(ENCARG(cons_b(cons_a(c(y0))))) ENCARG(c(cons_b(cons_a(cons_a(y0))))) -> c1(ENCARG(cons_b(cons_a(cons_a(y0))))) ENCARG(c(cons_b(cons_a(cons_b(y0))))) -> c1(ENCARG(cons_b(cons_a(cons_b(y0))))) ENCARG(c(cons_b(cons_b(cons_a(y0))))) -> c1(ENCARG(cons_b(cons_b(cons_a(y0))))) ENCARG(c(cons_b(cons_b(cons_b(y0))))) -> c1(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(c(cons_b(cons_b(c(y0))))) -> c1(ENCARG(cons_b(cons_b(c(y0))))) ENCARG(cons_a(cons_a(z0))) -> c2(ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(ENCARG(cons_b(cons_b(z0)))) B(a(c(x0))) -> c8 B(a(c(c(z0)))) -> c8 B(a(c(a(z0)))) -> c8 S tuples: B(a(c(x0))) -> c8 B(a(c(c(z0)))) -> c8 B(a(c(a(z0)))) -> c8 K tuples:none Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, ENCODE_B_1, B_1 Compound Symbols: c2_1, c3_1, c3_2, c6_1, c1_1, c8 ---------------------------------------- (71) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. B(a(c(x0))) -> c8 B(a(c(c(z0)))) -> c8 B(a(c(a(z0)))) -> c8 We considered the (Usable) Rules: a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) And the Tuples: ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) ENCODE_B(cons_a(c(z0))) -> c6(B(a(c(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c6(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c6(B(a(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c6(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c6(B(b(b(encArg(z0))))) ENCARG(c(c(y0))) -> c1(ENCARG(c(y0))) ENCARG(c(cons_a(cons_a(y0)))) -> c1(ENCARG(cons_a(cons_a(y0)))) ENCARG(c(cons_a(c(y0)))) -> c1(ENCARG(cons_a(c(y0)))) ENCARG(c(cons_b(c(y0)))) -> c1(ENCARG(cons_b(c(y0)))) ENCARG(c(cons_a(cons_b(cons_a(y0))))) -> c1(ENCARG(cons_a(cons_b(cons_a(y0))))) ENCARG(c(cons_a(cons_b(cons_b(y0))))) -> c1(ENCARG(cons_a(cons_b(cons_b(y0))))) ENCARG(c(cons_a(cons_b(c(y0))))) -> c1(ENCARG(cons_a(cons_b(c(y0))))) ENCARG(c(cons_b(cons_a(c(y0))))) -> c1(ENCARG(cons_b(cons_a(c(y0))))) ENCARG(c(cons_b(cons_a(cons_a(y0))))) -> c1(ENCARG(cons_b(cons_a(cons_a(y0))))) ENCARG(c(cons_b(cons_a(cons_b(y0))))) -> c1(ENCARG(cons_b(cons_a(cons_b(y0))))) ENCARG(c(cons_b(cons_b(cons_a(y0))))) -> c1(ENCARG(cons_b(cons_b(cons_a(y0))))) ENCARG(c(cons_b(cons_b(cons_b(y0))))) -> c1(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(c(cons_b(cons_b(c(y0))))) -> c1(ENCARG(cons_b(cons_b(c(y0))))) ENCARG(cons_a(cons_a(z0))) -> c2(ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(ENCARG(cons_b(cons_b(z0)))) B(a(c(x0))) -> c8 B(a(c(c(z0)))) -> c8 B(a(c(a(z0)))) -> c8 The order we found is given by the following interpretation: Polynomial interpretation : POL(B(x_1)) = x_1 POL(ENCARG(x_1)) = x_1 POL(ENCODE_B(x_1)) = [1] POL(a(x_1)) = [1] POL(b(x_1)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8) = 0 POL(cons_a(x_1)) = [1] + x_1 POL(cons_b(x_1)) = [1] + x_1 POL(encArg(x_1)) = 0 ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_b(z0)) -> b(encArg(z0)) a(a(z0)) -> b(z0) b(a(c(z0))) -> c(c(a(a(a(a(z0)))))) Tuples: ENCARG(cons_a(c(z0))) -> c2(ENCARG(c(z0))) ENCARG(cons_b(c(z0))) -> c3(ENCARG(c(z0))) ENCARG(cons_a(cons_b(c(z0)))) -> c2(ENCARG(cons_b(c(z0)))) ENCARG(cons_b(cons_a(c(z0)))) -> c3(B(a(c(encArg(z0)))), ENCARG(cons_a(c(z0)))) ENCARG(cons_b(cons_a(cons_a(z0)))) -> c3(B(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_b(cons_a(cons_b(z0)))) -> c3(B(a(b(encArg(z0)))), ENCARG(cons_a(cons_b(z0)))) ENCARG(cons_b(cons_b(cons_a(z0)))) -> c3(B(b(a(encArg(z0)))), ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_b(cons_b(cons_b(z0)))) -> c3(B(b(b(encArg(z0)))), ENCARG(cons_b(cons_b(z0)))) ENCARG(cons_b(cons_b(c(z0)))) -> c3(ENCARG(cons_b(c(z0)))) ENCODE_B(cons_a(c(z0))) -> c6(B(a(c(encArg(z0))))) ENCODE_B(cons_a(cons_a(z0))) -> c6(B(a(a(encArg(z0))))) ENCODE_B(cons_a(cons_b(z0))) -> c6(B(a(b(encArg(z0))))) ENCODE_B(cons_b(cons_a(z0))) -> c6(B(b(a(encArg(z0))))) ENCODE_B(cons_b(cons_b(z0))) -> c6(B(b(b(encArg(z0))))) ENCARG(c(c(y0))) -> c1(ENCARG(c(y0))) ENCARG(c(cons_a(cons_a(y0)))) -> c1(ENCARG(cons_a(cons_a(y0)))) ENCARG(c(cons_a(c(y0)))) -> c1(ENCARG(cons_a(c(y0)))) ENCARG(c(cons_b(c(y0)))) -> c1(ENCARG(cons_b(c(y0)))) ENCARG(c(cons_a(cons_b(cons_a(y0))))) -> c1(ENCARG(cons_a(cons_b(cons_a(y0))))) ENCARG(c(cons_a(cons_b(cons_b(y0))))) -> c1(ENCARG(cons_a(cons_b(cons_b(y0))))) ENCARG(c(cons_a(cons_b(c(y0))))) -> c1(ENCARG(cons_a(cons_b(c(y0))))) ENCARG(c(cons_b(cons_a(c(y0))))) -> c1(ENCARG(cons_b(cons_a(c(y0))))) ENCARG(c(cons_b(cons_a(cons_a(y0))))) -> c1(ENCARG(cons_b(cons_a(cons_a(y0))))) ENCARG(c(cons_b(cons_a(cons_b(y0))))) -> c1(ENCARG(cons_b(cons_a(cons_b(y0))))) ENCARG(c(cons_b(cons_b(cons_a(y0))))) -> c1(ENCARG(cons_b(cons_b(cons_a(y0))))) ENCARG(c(cons_b(cons_b(cons_b(y0))))) -> c1(ENCARG(cons_b(cons_b(cons_b(y0))))) ENCARG(c(cons_b(cons_b(c(y0))))) -> c1(ENCARG(cons_b(cons_b(c(y0))))) ENCARG(cons_a(cons_a(z0))) -> c2(ENCARG(cons_a(z0))) ENCARG(cons_a(cons_b(cons_a(z0)))) -> c2(ENCARG(cons_b(cons_a(z0)))) ENCARG(cons_a(cons_b(cons_b(z0)))) -> c2(ENCARG(cons_b(cons_b(z0)))) B(a(c(x0))) -> c8 B(a(c(c(z0)))) -> c8 B(a(c(a(z0)))) -> c8 S tuples:none K tuples: B(a(c(x0))) -> c8 B(a(c(c(z0)))) -> c8 B(a(c(a(z0)))) -> c8 Defined Rule Symbols: encArg_1, a_1, b_1 Defined Pair Symbols: ENCARG_1, ENCODE_B_1, B_1 Compound Symbols: c2_1, c3_1, c3_2, c6_1, c1_1, c8 ---------------------------------------- (73) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (74) BOUNDS(1, 1)