/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, 1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 166 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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CdtProblem (23) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 132 ms] (58) CdtProblem (59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (64) BOUNDS(1, 1) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, 1). The TRS R consists of the following rules: f(f(x, a), y) -> f(y, f(x, 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, 1). The TRS R consists of the following rules: f(f(x, a), y) -> f(y, f(x, 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, 1). The TRS R consists of the following rules: f(f(x, a), y) -> f(y, f(x, 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(f(z0, a), z1) -> f(z1, f(z0, z1)) 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(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) S tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) 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_2 ---------------------------------------- (7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: ENCODE_A -> c3 ENCARG(a) -> c ---------------------------------------- (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(f(z0, a), z1) -> f(z1, f(z0, z1)) 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(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) S tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) 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_2 ---------------------------------------- (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(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) 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(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) 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_2, 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(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) S tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) 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_2, 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(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) S tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) 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_2, 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(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) 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(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCODE_F_2, ENCARG_1 Compound Symbols: c4_2, 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(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) 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(F(encArg(x0), a), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) S tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCODE_F_2, ENCARG_1 Compound Symbols: c4_2, c_1, c1_3, c1_2, c1_1 ---------------------------------------- (19) 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))) ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, 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(F(encArg(x0), a), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) 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(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_3, c1_2, c1_1, c_1 ---------------------------------------- (21) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, 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(F(encArg(x0), a), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) 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(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) S tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_3, c1_2, c1_1, c_1 ---------------------------------------- (23) 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))) ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), 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(F(encArg(x0), a), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) 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(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(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_3, c1_2, c1_1, c_1 ---------------------------------------- (25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), 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(F(encArg(x0), a), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) 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(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(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_3, c1_2, c1_1, c_1 ---------------------------------------- (27) 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)) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) ENCARG(cons_f(x0, a)) -> c1(F(encArg(x0), a), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) 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(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(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_2, c1_1, c_1, c1_3 ---------------------------------------- (29) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) ENCARG(cons_f(x0, a)) -> c1(F(encArg(x0), a), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) 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(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(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) S tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_2, c1_1, c_1, c1_3 ---------------------------------------- (31) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(x0, a)) -> c1(F(encArg(x0), a), ENCARG(x0)) by ENCARG(cons_f(a, a)) -> c1(F(a, a), ENCARG(a)) ENCARG(cons_f(cons_f(z0, z1), a)) -> c1(F(f(encArg(z0), encArg(z1)), a), ENCARG(cons_f(z0, z1))) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) 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(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(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, a)) -> c1(F(a, a), ENCARG(a)) S tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c_1, c1_3, c1_2 ---------------------------------------- (33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCARG(cons_f(a, a)) -> c1(F(a, a), ENCARG(a)) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) 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(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(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) S tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c_1, c1_3, c1_2 ---------------------------------------- (35) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(x0, a) -> c(F(encArg(x0), a)) by ENCODE_F(a, a) -> c(F(a, a)) ENCODE_F(cons_f(z0, z1), a) -> c(F(f(encArg(z0), encArg(z1)), a)) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) 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(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCODE_F(a, a) -> c(F(a, a)) ENCODE_F(cons_f(z0, z1), a) -> c(F(f(encArg(z0), encArg(z1)), a)) S tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c_1, c1_3, c1_2 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_F(a, a) -> c(F(a, a)) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) 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(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c(F(f(encArg(z0), encArg(z1)), a)) S tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c_1, c1_3, c1_2 ---------------------------------------- (39) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) by ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) 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(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c(F(f(encArg(z0), encArg(z1)), a)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) S tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c_1, c1_3, c1_2 ---------------------------------------- (41) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) 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(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c(F(f(encArg(z0), encArg(z1)), a)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) S tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c_1, c1_3, c1_2 ---------------------------------------- (43) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) by ENCODE_F(cons_f(x0, x1), a) -> c(F(f(encArg(x0), encArg(x1)), a)) ENCODE_F(cons_f(x0, x1), cons_f(z0, z1)) -> c(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) ENCARG(cons_f(a, x1)) -> c1(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(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c(F(f(encArg(z0), encArg(z1)), a)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) S tuples: F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c1_3, c1_2, c_1 ---------------------------------------- (45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(f(z0, a), z1) -> c4(F(z1, f(z0, z1)), F(z0, z1)) by F(f(z0, a), f(y0, a)) -> c4(F(f(y0, a), f(z0, f(y0, a))), F(z0, f(y0, a))) F(f(f(y0, a), a), z1) -> c4(F(z1, f(f(y0, a), z1)), F(f(y0, a), z1)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: ENCARG(cons_f(a, x1)) -> c1(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(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c(F(f(encArg(z0), encArg(z1)), a)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(f(z0, a), f(y0, a)) -> c4(F(f(y0, a), f(z0, f(y0, a))), F(z0, f(y0, a))) F(f(f(y0, a), a), z1) -> c4(F(z1, f(f(y0, a), z1)), F(f(y0, a), z1)) S tuples: F(f(z0, a), f(y0, a)) -> c4(F(f(y0, a), f(z0, f(y0, a))), F(z0, f(y0, a))) F(f(f(y0, a), a), z1) -> c4(F(z1, f(f(y0, a), z1)), F(f(y0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c1_3, c1_2, c_1, c4_2 ---------------------------------------- (47) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: F(f(f(y0, a), a), z1) -> c4(F(z1, f(f(y0, a), z1)), F(f(y0, a), z1)) Removed 1 trailing nodes: ENCODE_F(cons_f(x0, x1), a) -> c(F(f(encArg(x0), encArg(x1)), a)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: ENCARG(cons_f(a, x1)) -> c1(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(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(f(z0, a), f(y0, a)) -> c4(F(f(y0, a), f(z0, f(y0, a))), F(z0, f(y0, a))) S tuples: F(f(z0, a), f(y0, a)) -> c4(F(f(y0, a), f(z0, f(y0, a))), F(z0, f(y0, 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, c1_3, c1_2, c_1, c4_2 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: ENCARG(cons_f(a, x1)) -> c1(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)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_f(cons_f(x0, x1), a)) -> c1(ENCARG(cons_f(x0, x1))) F(f(z0, a), f(y0, a)) -> c4(F(z0, f(y0, a))) S tuples: F(f(z0, a), f(y0, a)) -> c4(F(z0, f(y0, 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, c1_3, c_1, c4_1 ---------------------------------------- (51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_f(a, x1)) -> c1(ENCARG(x1)) by ENCARG(cons_f(a, cons_f(a, y0))) -> c1(ENCARG(cons_f(a, y0))) ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1))) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1))) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: 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)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_f(cons_f(x0, x1), a)) -> c1(ENCARG(cons_f(x0, x1))) F(f(z0, a), f(y0, a)) -> c4(F(z0, f(y0, a))) ENCARG(cons_f(a, cons_f(a, y0))) -> c1(ENCARG(cons_f(a, y0))) ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1))) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1))) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) S tuples: F(f(z0, a), f(y0, a)) -> c4(F(z0, f(y0, a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_1, c_1, c4_1 ---------------------------------------- (53) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_f(a, cons_f(x1, x2))) -> c1(ENCARG(cons_f(x1, x2))) by ENCARG(cons_f(a, cons_f(z0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(z0, cons_f(y1, a)))) ENCARG(cons_f(a, cons_f(z0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(z0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(a, cons_f(z0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(z0, cons_f(a, y1)))) ENCARG(cons_f(a, cons_f(z0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(z0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(a, cons_f(cons_f(y0, a), z1))) -> c1(ENCARG(cons_f(cons_f(y0, a), z1))) ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), z1))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), z1))) ENCARG(cons_f(a, cons_f(cons_f(a, y0), z1))) -> c1(ENCARG(cons_f(cons_f(a, y0), z1))) ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), z1))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), z1))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(cons_f(a, cons_f(a, cons_f(a, y0)))) -> c1(ENCARG(cons_f(a, cons_f(a, y0)))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) ENCARG(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1))))) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1))))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a)))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: 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(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)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_f(cons_f(x0, x1), a)) -> c1(ENCARG(cons_f(x0, x1))) F(f(z0, a), f(y0, a)) -> c4(F(z0, f(y0, a))) ENCARG(cons_f(a, cons_f(a, y0))) -> c1(ENCARG(cons_f(a, y0))) ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1))) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1))) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(cons_f(a, cons_f(a, cons_f(a, y0)))) -> c1(ENCARG(cons_f(a, cons_f(a, y0)))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) ENCARG(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1))))) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1))))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a)))) S tuples: F(f(z0, a), f(y0, a)) -> c4(F(z0, f(y0, a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c_1, c1_1, c4_1 ---------------------------------------- (55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_f(cons_f(x0, x1), a)) -> c1(ENCARG(cons_f(x0, x1))) by ENCARG(cons_f(cons_f(z0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(z0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(z0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(z0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(z0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(z0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(z0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(z0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(cons_f(y0, a), z1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), z1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), z1), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), z1))) ENCARG(cons_f(cons_f(cons_f(a, y0), z1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), z1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), z1), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), z1))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(cons_f(cons_f(a, cons_f(a, y0)), a)) -> c1(ENCARG(cons_f(a, cons_f(a, y0)))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, a))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(a, y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1))))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, a), y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(a, y0), y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), a)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a)))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, y0))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(a, y0))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: 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(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)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_f(cons_f(x0, x1), a)) -> c1(ENCARG(cons_f(x0, x1))) F(f(z0, a), f(y0, a)) -> c4(F(z0, f(y0, a))) ENCARG(cons_f(a, cons_f(a, y0))) -> c1(ENCARG(cons_f(a, y0))) ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1))) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1))) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(cons_f(a, cons_f(a, cons_f(a, y0)))) -> c1(ENCARG(cons_f(a, cons_f(a, y0)))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) ENCARG(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1))))) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1))))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a)))) ENCARG(cons_f(cons_f(z0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(z0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(z0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(z0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(z0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(z0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(z0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(z0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(cons_f(y0, a), z1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), z1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), z1), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), z1))) ENCARG(cons_f(cons_f(cons_f(a, y0), z1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), z1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), z1), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), z1))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(cons_f(cons_f(a, cons_f(a, y0)), a)) -> c1(ENCARG(cons_f(a, cons_f(a, y0)))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, a))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(a, y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1))))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, a), y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(a, y0), y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), a)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a)))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, y0))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(a, y0))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))))) S tuples: F(f(z0, a), f(y0, a)) -> c4(F(z0, f(y0, a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c_1, c1_1, c4_1 ---------------------------------------- (57) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_f(cons_f(z0, z1), a)) -> c1(ENCARG(cons_f(z0, z1))) by ENCARG(cons_f(cons_f(z0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(z0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(z0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(z0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(z0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(z0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(z0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(z0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(cons_f(y0, a), z1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), z1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), z1), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), z1))) ENCARG(cons_f(cons_f(cons_f(a, y0), z1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), z1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), z1), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), z1))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(cons_f(cons_f(a, cons_f(a, y0)), a)) -> c1(ENCARG(cons_f(a, cons_f(a, y0)))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, a))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(a, y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1))))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, a), y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(a, y0), y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), a)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a)))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, y0))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(a, y0))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, y0)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, y0)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(y1, a))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, a))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(a, y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(a, y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, a), y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, a), y1)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(a, y0), y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(a, y0), y1)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, y1), a)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), a)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(a, y0))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, y0))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))), a))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: 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(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)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(f(z0, a), f(y0, a)) -> c4(F(z0, f(y0, a))) ENCARG(cons_f(a, cons_f(a, y0))) -> c1(ENCARG(cons_f(a, y0))) ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1))) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1))) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(cons_f(a, cons_f(a, cons_f(a, y0)))) -> c1(ENCARG(cons_f(a, cons_f(a, y0)))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) ENCARG(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1))))) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1))))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a)))) ENCARG(cons_f(cons_f(z0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(z0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(z0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(z0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(z0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(z0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(z0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(z0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(cons_f(y0, a), z1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), z1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), z1), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), z1))) ENCARG(cons_f(cons_f(cons_f(a, y0), z1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), z1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), z1), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), z1))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(cons_f(cons_f(a, cons_f(a, y0)), a)) -> c1(ENCARG(cons_f(a, cons_f(a, y0)))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, a))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(a, y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1))))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, a), y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(a, y0), y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), a)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a)))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, y0))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(a, y0))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, y0)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, y0)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(y1, a))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, a))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(a, y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(a, y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, a), y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, a), y1)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(a, y0), y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(a, y0), y1)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, y1), a)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), a)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(a, y0))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, y0))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))), a))) S tuples: F(f(z0, a), f(y0, a)) -> c4(F(z0, f(y0, a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c_1, c4_1, c1_1 ---------------------------------------- (59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(f(z0, a), f(y0, a)) -> c4(F(z0, f(y0, a))) by F(f(f(y0, a), a), f(z1, a)) -> c4(F(f(y0, a), f(z1, a))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: 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(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)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_f(a, cons_f(a, y0))) -> c1(ENCARG(cons_f(a, y0))) ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1))) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1))) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(cons_f(a, cons_f(a, cons_f(a, y0)))) -> c1(ENCARG(cons_f(a, cons_f(a, y0)))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) ENCARG(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1))))) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1))))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a)))) ENCARG(cons_f(cons_f(z0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(z0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(z0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(z0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(z0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(z0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(z0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(z0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(cons_f(y0, a), z1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), z1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), z1), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), z1))) ENCARG(cons_f(cons_f(cons_f(a, y0), z1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), z1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), z1), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), z1))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(cons_f(cons_f(a, cons_f(a, y0)), a)) -> c1(ENCARG(cons_f(a, cons_f(a, y0)))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, a))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(a, y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1))))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, a), y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(a, y0), y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), a)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a)))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, y0))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(a, y0))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, y0)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, y0)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(y1, a))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, a))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(a, y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(a, y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, a), y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, a), y1)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(a, y0), y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(a, y0), y1)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, y1), a)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), a)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(a, y0))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, y0))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))), a))) F(f(f(y0, a), a), f(z1, a)) -> c4(F(f(y0, a), f(z1, a))) S tuples: F(f(f(y0, a), a), f(z1, a)) -> c4(F(f(y0, a), f(z1, a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c_1, c1_1, c4_1 ---------------------------------------- (61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCODE_F(cons_f(x0, x1), cons_f(z0, z1)) -> c(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) F(f(f(y0, a), a), f(z1, a)) -> c4(F(f(y0, a), f(z1, a))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(z0, a), z1) -> f(z1, f(z0, z1)) Tuples: 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(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(a, cons_f(a, y0))) -> c1(ENCARG(cons_f(a, y0))) ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1))) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1))) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(cons_f(a, cons_f(a, cons_f(a, y0)))) -> c1(ENCARG(cons_f(a, cons_f(a, y0)))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1))))) ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) ENCARG(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1))))) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1))))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a)))) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a)))) ENCARG(cons_f(cons_f(z0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(z0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(z0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(z0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(z0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(z0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(z0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(z0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(cons_f(y0, a), z1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), z1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), z1), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), z1))) ENCARG(cons_f(cons_f(cons_f(a, y0), z1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), z1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), z1), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), z1))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(cons_f(cons_f(a, cons_f(a, y0)), a)) -> c1(ENCARG(cons_f(a, cons_f(a, y0)))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, a))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, a))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(a, y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(a, y1))))) ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, y1))))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, a), y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, a), y1)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(a, y0), y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(a, y0), y1)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)))) ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), a)), a)) -> c1(ENCARG(cons_f(a, cons_f(cons_f(y0, y1), a)))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, y0))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(a, y0))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))))) ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))), a)) -> c1(ENCARG(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, y0)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, y0)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(y1, a))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, a))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(a, y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(a, y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, a), y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, a), y1)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(a, y0), y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(a, y0), y1)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(cons_f(y0, y1), a)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(cons_f(y0, y1), a)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(a, y0))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, y0))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, a)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(y1, cons_f(y2, y3))))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(a, y1)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(y0, cons_f(cons_f(y1, y2), y3)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), cons_f(y2, y3)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(a, cons_f(y0, y1)))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, a), y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, cons_f(y1, y2)), y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(a, y0), y1))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(cons_f(y0, y1), y2), y3))), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(a, cons_f(cons_f(y0, y1), a))), a))) S tuples:none K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1 Compound Symbols: c1_3, c1_1 ---------------------------------------- (63) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (64) BOUNDS(1, 1)