/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 161 ms] (4) CpxRelTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (6) CdtProblem (7) CdtLeafRemovalProof [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) 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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 23 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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (34) CdtProblem (35) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (38) CdtProblem (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms] (42) CdtProblem (43) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (44) 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: f(y, f(y, x)) -> f(f(a, y), f(a, y)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(a) -> a encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_a -> a ---------------------------------------- (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: f(y, f(y, x)) -> f(f(a, y), f(a, y)) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_a -> a 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: f(y, f(y, x)) -> f(f(a, y), f(a, y)) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_a -> a Rewrite Strategy: INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCARG(a) -> c ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_F(z0, z1) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_A -> c3 F(z0, f(z0, z1)) -> c4(F(f(a, z0), f(a, z0)), F(a, z0), F(a, z0)) S tuples: F(z0, f(z0, z1)) -> c4(F(f(a, z0), f(a, z0)), F(a, z0), F(a, z0)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a Defined Pair Symbols: ENCARG_1, ENCODE_F_2, ENCODE_A, F_2 Compound Symbols: c, c1_3, c2_3, c3, c4_3 ---------------------------------------- (7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: ENCARG(a) -> c ENCODE_A -> c3 ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_F(z0, z1) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(z0, f(z0, z1)) -> c4(F(f(a, z0), f(a, z0)), F(a, z0), F(a, z0)) S tuples: F(z0, f(z0, z1)) -> c4(F(f(a, z0), f(a, z0)), F(a, z0), F(a, z0)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c2_3, c4_3 ---------------------------------------- (9) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(z0, f(z0, z1)) -> c4(F(f(a, z0), f(a, z0)), F(a, z0), F(a, z0)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCODE_F(z0, z1) -> c(ENCARG(z0)) ENCODE_F(z0, z1) -> c(ENCARG(z1)) S tuples: F(z0, f(z0, z1)) -> c4(F(f(a, z0), f(a, z0)), F(a, z0), F(a, z0)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_3, c4_3, c_1 ---------------------------------------- (11) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ENCODE_F(z0, z1) -> c(ENCARG(z0)) ENCODE_F(z0, z1) -> c(ENCARG(z1)) ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(z0, f(z0, z1)) -> c4(F(f(a, z0), f(a, z0)), F(a, z0), F(a, z0)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) S tuples: F(z0, f(z0, z1)) -> c4(F(f(a, z0), f(a, z0)), F(a, z0), F(a, z0)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_3, c4_3, c_1 ---------------------------------------- (13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(z0, f(z0, z1)) -> c4(F(f(a, z0), f(a, z0)), F(a, z0), F(a, z0)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) S tuples: F(z0, f(z0, z1)) -> c4(F(f(a, z0), f(a, z0)), F(a, z0), F(a, z0)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_3, c4_3, c_1 ---------------------------------------- (15) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by ENCARG(cons_f(x0, a)) -> c1(F(encArg(x0), a), ENCARG(x0), ENCARG(a)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(a), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: F(z0, f(z0, z1)) -> c4(F(f(a, z0), f(a, z0)), F(a, z0), F(a, z0)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_f(x0, a)) -> c1(F(encArg(x0), a), ENCARG(x0), ENCARG(a)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(a), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) S tuples: F(z0, f(z0, z1)) -> c4(F(f(a, z0), f(a, z0)), F(a, z0), F(a, z0)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCODE_F_2, ENCARG_1 Compound Symbols: c4_3, c_1, c1_3 ---------------------------------------- (17) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: F(z0, f(z0, z1)) -> c4(F(f(a, z0), f(a, z0)), F(a, z0), F(a, z0)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(x1)) S tuples: F(z0, f(z0, z1)) -> c4(F(f(a, z0), f(a, z0)), F(a, z0), F(a, z0)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCODE_F_2, ENCARG_1 Compound Symbols: c4_3, c_1, c1_3, c1_1, c1_2 ---------------------------------------- (19) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(z0, f(z0, z1)) -> c4(F(f(a, z0), f(a, z0)), F(a, z0), F(a, z0)) by F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a))), F(a, f(a, z1)), F(a, f(a, z1))) F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(f(a, a), f(a, a)), f(a, f(a, z1))), F(a, f(a, z1)), F(a, f(a, z1))) ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(x1)) F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a))), F(a, f(a, z1)), F(a, f(a, z1))) F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(f(a, a), f(a, a)), f(a, f(a, z1))), F(a, f(a, z1)), F(a, f(a, z1))) S tuples: F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a))), F(a, f(a, z1)), F(a, f(a, z1))) F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(f(a, a), f(a, a)), f(a, f(a, z1))), F(a, f(a, z1)), F(a, f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCODE_F_2, ENCARG_1, F_2 Compound Symbols: c_1, c1_3, c1_1, c1_2, c4_3 ---------------------------------------- (21) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) F(f(a, z1), f(f(a, z1), x1)) -> c4 S tuples: F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) F(f(a, z1), f(f(a, z1), x1)) -> c4 K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCODE_F_2, ENCARG_1, F_2 Compound Symbols: c_1, c1_3, c1_1, c4_1, c4 ---------------------------------------- (23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(f(a, z1), f(f(a, z1), x1)) -> c4 We considered the (Usable) Rules:none And the Tuples: ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) F(f(a, z1), f(f(a, z1), x1)) -> c4 The order we found is given by the following interpretation: Polynomial interpretation : POL(ENCARG(x_1)) = x_1 POL(ENCODE_F(x_1, x_2)) = [1] POL(F(x_1, x_2)) = [1] POL(a) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4) = 0 POL(c4(x_1)) = x_1 POL(cons_f(x_1, x_2)) = [1] + x_1 + x_2 POL(encArg(x_1)) = [1] + x_1 POL(f(x_1, x_2)) = 0 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) F(f(a, z1), f(f(a, z1), x1)) -> c4 S tuples: F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) K tuples: F(f(a, z1), f(f(a, z1), x1)) -> c4 Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCODE_F_2, ENCARG_1, F_2 Compound Symbols: c_1, c1_3, c1_1, c4_1, c4 ---------------------------------------- (25) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) by ENCODE_F(x0, a) -> c(F(encArg(x0), a)) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) F(f(a, z1), f(f(a, z1), x1)) -> c4 ENCODE_F(x0, a) -> c(F(encArg(x0), a)) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) S tuples: F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) K tuples: F(f(a, z1), f(f(a, z1), x1)) -> c4 Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_3, c1_1, c4_1, c4, c_1 ---------------------------------------- (27) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: F(f(a, z1), f(f(a, z1), x1)) -> c4 ENCODE_F(x0, a) -> c(F(encArg(x0), a)) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) S tuples: F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_3, c1_1, c4_1, c_1 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) by ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(a), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(a), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) S tuples: F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_3, c1_1, c4_1, c_1 ---------------------------------------- (31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c1(ENCARG(cons_f(x1, x2))) S tuples: F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_3, c1_1, c4_1, c_1 ---------------------------------------- (33) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) by ENCARG(cons_f(cons_f(x0, x1), a)) -> c1(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1)), ENCARG(a)) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c1(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c1(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, x1), a)) -> c1(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1)), ENCARG(a)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) S tuples: F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c4_1, c_1, c1_3 ---------------------------------------- (35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c1(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c1(ENCARG(cons_f(x0, x1))) S tuples: F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c4_1, c_1, c1_3 ---------------------------------------- (37) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(a, f(a, z1)), f(f(a, a), f(a, a)))) by F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(f(a, a), f(a, a)), f(f(a, a), f(a, a)))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c1(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c1(ENCARG(cons_f(x0, x1))) F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(f(a, a), f(a, a)), f(f(a, a), f(a, a)))) S tuples: F(f(a, z1), f(f(a, z1), x1)) -> c4(F(f(f(a, a), f(a, a)), f(f(a, a), f(a, a)))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c_1, c1_3, c4_1 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c1(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c1(ENCARG(cons_f(x0, x1))) F(f(a, z1), f(f(a, z1), x1)) -> c4 S tuples: F(f(a, z1), f(f(a, z1), x1)) -> c4 K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c_1, c1_3, c4 ---------------------------------------- (41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(f(a, z1), f(f(a, z1), x1)) -> c4 We considered the (Usable) Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) And the Tuples: ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c1(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c1(ENCARG(cons_f(x0, x1))) F(f(a, z1), f(f(a, z1), x1)) -> c4 The order we found is given by the following interpretation: Polynomial interpretation : POL(ENCARG(x_1)) = x_1 POL(ENCODE_F(x_1, x_2)) = [1] + x_1 + x_2 POL(F(x_1, x_2)) = x_1 POL(a) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4) = 0 POL(cons_f(x_1, x_2)) = [1] + x_1 + x_2 POL(encArg(x_1)) = [1] POL(f(x_1, x_2)) = [1] ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z0, z1)) -> f(f(a, z0), f(a, z0)) Tuples: ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c1(ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c1(ENCARG(cons_f(x0, x1))) F(f(a, z1), f(f(a, z1), x1)) -> c4 S tuples:none K tuples: F(f(a, z1), f(f(a, z1), x1)) -> c4 Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c_1, c1_3, c4 ---------------------------------------- (43) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (44) BOUNDS(1, 1)