/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 324 ms] (4) CpxRelTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (6) CdtProblem (7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (8) CdtProblem (9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (12) CdtProblem (13) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CdtProblem (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 199 ms] (16) CdtProblem (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 736 ms] (18) CdtProblem (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 610 ms] (20) CdtProblem (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 585 ms] (22) CdtProblem (23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 468 ms] (24) CdtProblem (25) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 527 ms] (26) CdtProblem (27) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 515 ms] (28) CdtProblem (29) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (30) BOUNDS(1, 1) (31) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (32) TRS for Loop Detection (33) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (34) BEST (35) proven lower bound (36) LowerBoundPropagationProof [FINISHED, 0 ms] (37) BOUNDS(n^1, INF) (38) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: minus(X, s(Y)) -> pred(minus(X, Y)) minus(X, 0) -> X pred(s(X)) -> X le(s(X), s(Y)) -> le(X, Y) le(s(X), 0) -> false le(0, Y) -> true gcd(0, Y) -> 0 gcd(s(X), 0) -> s(X) gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) 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(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_pred(x_1)) -> pred(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_pred(x_1) -> pred(encArg(x_1)) encode_0 -> 0 encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_false -> false encode_true -> true encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: minus(X, s(Y)) -> pred(minus(X, Y)) minus(X, 0) -> X pred(s(X)) -> X le(s(X), s(Y)) -> le(X, Y) le(s(X), 0) -> false le(0, Y) -> true gcd(0, Y) -> 0 gcd(s(X), 0) -> s(X) gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_pred(x_1)) -> pred(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_pred(x_1) -> pred(encArg(x_1)) encode_0 -> 0 encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_false -> false encode_true -> true encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) 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(n^1, n^2). The TRS R consists of the following rules: minus(X, s(Y)) -> pred(minus(X, Y)) minus(X, 0) -> X pred(s(X)) -> X le(s(X), s(Y)) -> le(X, Y) le(s(X), 0) -> false le(0, Y) -> true gcd(0, Y) -> 0 gcd(s(X), 0) -> s(X) gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_pred(x_1)) -> pred(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_pred(x_1) -> pred(encArg(x_1)) encode_0 -> 0 encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_false -> false encode_true -> true encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) encArg(cons_pred(z0)) -> pred(encArg(z0)) encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) encode_minus(z0, z1) -> minus(encArg(z0), encArg(z1)) encode_s(z0) -> s(encArg(z0)) encode_pred(z0) -> pred(encArg(z0)) encode_0 -> 0 encode_le(z0, z1) -> le(encArg(z0), encArg(z1)) encode_false -> false encode_true -> true encode_gcd(z0, z1) -> gcd(encArg(z0), encArg(z1)) encode_if(z0, z1, z2) -> if(encArg(z0), encArg(z1), encArg(z2)) minus(z0, s(z1)) -> pred(minus(z0, z1)) minus(z0, 0) -> z0 pred(s(z0)) -> z0 le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true gcd(0, z0) -> 0 gcd(s(z0), 0) -> s(z0) gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(0) -> c1 ENCARG(false) -> c2 ENCARG(true) -> c3 ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) ENCODE_MINUS(z0, z1) -> c9(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_S(z0) -> c10(ENCARG(z0)) ENCODE_PRED(z0) -> c11(PRED(encArg(z0)), ENCARG(z0)) ENCODE_0 -> c12 ENCODE_LE(z0, z1) -> c13(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_FALSE -> c14 ENCODE_TRUE -> c15 ENCODE_GCD(z0, z1) -> c16(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_IF(z0, z1, z2) -> c17(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) S tuples: MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) K tuples:none Defined Rule Symbols: minus_2, pred_1, le_2, gcd_2, if_3, encArg_1, encode_minus_2, encode_s_1, encode_pred_1, encode_0, encode_le_2, encode_false, encode_true, encode_gcd_2, encode_if_3 Defined Pair Symbols: ENCARG_1, ENCODE_MINUS_2, ENCODE_S_1, ENCODE_PRED_1, ENCODE_0, ENCODE_LE_2, ENCODE_FALSE, ENCODE_TRUE, ENCODE_GCD_2, ENCODE_IF_3, MINUS_2, PRED_1, LE_2, GCD_2, IF_3 Compound Symbols: c_1, c1, c2, c3, c4_3, c5_2, c6_3, c7_3, c8_4, c9_3, c10_1, c11_2, c12, c13_3, c14, c15, c16_3, c17_4, c18_2, c19, c20, c21_1, c22, c23, c24, c25, c26_2, c27_2, c28_2 ---------------------------------------- (7) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ENCODE_S(z0) -> c10(ENCARG(z0)) Removed 6 trailing nodes: ENCARG(true) -> c3 ENCODE_TRUE -> c15 ENCARG(false) -> c2 ENCODE_FALSE -> c14 ENCODE_0 -> c12 ENCARG(0) -> c1 ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) encArg(cons_pred(z0)) -> pred(encArg(z0)) encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) encode_minus(z0, z1) -> minus(encArg(z0), encArg(z1)) encode_s(z0) -> s(encArg(z0)) encode_pred(z0) -> pred(encArg(z0)) encode_0 -> 0 encode_le(z0, z1) -> le(encArg(z0), encArg(z1)) encode_false -> false encode_true -> true encode_gcd(z0, z1) -> gcd(encArg(z0), encArg(z1)) encode_if(z0, z1, z2) -> if(encArg(z0), encArg(z1), encArg(z2)) minus(z0, s(z1)) -> pred(minus(z0, z1)) minus(z0, 0) -> z0 pred(s(z0)) -> z0 le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true gcd(0, z0) -> 0 gcd(s(z0), 0) -> s(z0) gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) ENCODE_MINUS(z0, z1) -> c9(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_PRED(z0) -> c11(PRED(encArg(z0)), ENCARG(z0)) ENCODE_LE(z0, z1) -> c13(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_GCD(z0, z1) -> c16(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_IF(z0, z1, z2) -> c17(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) S tuples: MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) K tuples:none Defined Rule Symbols: minus_2, pred_1, le_2, gcd_2, if_3, encArg_1, encode_minus_2, encode_s_1, encode_pred_1, encode_0, encode_le_2, encode_false, encode_true, encode_gcd_2, encode_if_3 Defined Pair Symbols: ENCARG_1, ENCODE_MINUS_2, ENCODE_PRED_1, ENCODE_LE_2, ENCODE_GCD_2, ENCODE_IF_3, MINUS_2, PRED_1, LE_2, GCD_2, IF_3 Compound Symbols: c_1, c4_3, c5_2, c6_3, c7_3, c8_4, c9_3, c11_2, c13_3, c16_3, c17_4, c18_2, c19, c20, c21_1, c22, c23, c24, c25, c26_2, c27_2, c28_2 ---------------------------------------- (9) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) encArg(cons_pred(z0)) -> pred(encArg(z0)) encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) encode_minus(z0, z1) -> minus(encArg(z0), encArg(z1)) encode_s(z0) -> s(encArg(z0)) encode_pred(z0) -> pred(encArg(z0)) encode_0 -> 0 encode_le(z0, z1) -> le(encArg(z0), encArg(z1)) encode_false -> false encode_true -> true encode_gcd(z0, z1) -> gcd(encArg(z0), encArg(z1)) encode_if(z0, z1, z2) -> if(encArg(z0), encArg(z1), encArg(z2)) minus(z0, s(z1)) -> pred(minus(z0, z1)) minus(z0, 0) -> z0 pred(s(z0)) -> z0 le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true gcd(0, z0) -> 0 gcd(s(z0), 0) -> s(z0) gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_MINUS(z0, z1) -> c1(ENCARG(z0)) ENCODE_MINUS(z0, z1) -> c1(ENCARG(z1)) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_PRED(z0) -> c1(ENCARG(z0)) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_LE(z0, z1) -> c1(ENCARG(z0)) ENCODE_LE(z0, z1) -> c1(ENCARG(z1)) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(ENCARG(z0)) ENCODE_GCD(z0, z1) -> c1(ENCARG(z1)) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) ENCODE_IF(z0, z1, z2) -> c1(ENCARG(z0)) ENCODE_IF(z0, z1, z2) -> c1(ENCARG(z1)) ENCODE_IF(z0, z1, z2) -> c1(ENCARG(z2)) S tuples: MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) K tuples:none Defined Rule Symbols: minus_2, pred_1, le_2, gcd_2, if_3, encArg_1, encode_minus_2, encode_s_1, encode_pred_1, encode_0, encode_le_2, encode_false, encode_true, encode_gcd_2, encode_if_3 Defined Pair Symbols: ENCARG_1, MINUS_2, PRED_1, LE_2, GCD_2, IF_3, ENCODE_MINUS_2, ENCODE_PRED_1, ENCODE_LE_2, ENCODE_GCD_2, ENCODE_IF_3 Compound Symbols: c_1, c4_3, c5_2, c6_3, c7_3, c8_4, c18_2, c19, c20, c21_1, c22, c23, c24, c25, c26_2, c27_2, c28_2, c1_1 ---------------------------------------- (11) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 10 leading nodes: ENCODE_MINUS(z0, z1) -> c1(ENCARG(z0)) ENCODE_MINUS(z0, z1) -> c1(ENCARG(z1)) ENCODE_PRED(z0) -> c1(ENCARG(z0)) ENCODE_LE(z0, z1) -> c1(ENCARG(z0)) ENCODE_LE(z0, z1) -> c1(ENCARG(z1)) ENCODE_GCD(z0, z1) -> c1(ENCARG(z0)) ENCODE_GCD(z0, z1) -> c1(ENCARG(z1)) ENCODE_IF(z0, z1, z2) -> c1(ENCARG(z0)) ENCODE_IF(z0, z1, z2) -> c1(ENCARG(z1)) ENCODE_IF(z0, z1, z2) -> c1(ENCARG(z2)) ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) encArg(cons_pred(z0)) -> pred(encArg(z0)) encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) encode_minus(z0, z1) -> minus(encArg(z0), encArg(z1)) encode_s(z0) -> s(encArg(z0)) encode_pred(z0) -> pred(encArg(z0)) encode_0 -> 0 encode_le(z0, z1) -> le(encArg(z0), encArg(z1)) encode_false -> false encode_true -> true encode_gcd(z0, z1) -> gcd(encArg(z0), encArg(z1)) encode_if(z0, z1, z2) -> if(encArg(z0), encArg(z1), encArg(z2)) minus(z0, s(z1)) -> pred(minus(z0, z1)) minus(z0, 0) -> z0 pred(s(z0)) -> z0 le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true gcd(0, z0) -> 0 gcd(s(z0), 0) -> s(z0) gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) S tuples: MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) K tuples:none Defined Rule Symbols: minus_2, pred_1, le_2, gcd_2, if_3, encArg_1, encode_minus_2, encode_s_1, encode_pred_1, encode_0, encode_le_2, encode_false, encode_true, encode_gcd_2, encode_if_3 Defined Pair Symbols: ENCARG_1, MINUS_2, PRED_1, LE_2, GCD_2, IF_3, ENCODE_MINUS_2, ENCODE_PRED_1, ENCODE_LE_2, ENCODE_GCD_2, ENCODE_IF_3 Compound Symbols: c_1, c4_3, c5_2, c6_3, c7_3, c8_4, c18_2, c19, c20, c21_1, c22, c23, c24, c25, c26_2, c27_2, c28_2, c1_1 ---------------------------------------- (13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_minus(z0, z1) -> minus(encArg(z0), encArg(z1)) encode_s(z0) -> s(encArg(z0)) encode_pred(z0) -> pred(encArg(z0)) encode_0 -> 0 encode_le(z0, z1) -> le(encArg(z0), encArg(z1)) encode_false -> false encode_true -> true encode_gcd(z0, z1) -> gcd(encArg(z0), encArg(z1)) encode_if(z0, z1, z2) -> if(encArg(z0), encArg(z1), encArg(z2)) ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) encArg(cons_pred(z0)) -> pred(encArg(z0)) encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) minus(z0, s(z1)) -> pred(minus(z0, z1)) minus(z0, 0) -> z0 pred(s(z0)) -> z0 le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true gcd(0, z0) -> 0 gcd(s(z0), 0) -> s(z0) gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) S tuples: MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) K tuples:none Defined Rule Symbols: encArg_1, minus_2, pred_1, le_2, gcd_2, if_3 Defined Pair Symbols: ENCARG_1, MINUS_2, PRED_1, LE_2, GCD_2, IF_3, ENCODE_MINUS_2, ENCODE_PRED_1, ENCODE_LE_2, ENCODE_GCD_2, ENCODE_IF_3 Compound Symbols: c_1, c4_3, c5_2, c6_3, c7_3, c8_4, c18_2, c19, c20, c21_1, c22, c23, c24, c25, c26_2, c27_2, c28_2, c1_1 ---------------------------------------- (15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 We considered the (Usable) Rules:none And the Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ENCARG(x_1)) = x_1 POL(ENCODE_GCD(x_1, x_2)) = [1] + x_2 POL(ENCODE_IF(x_1, x_2, x_3)) = [1] POL(ENCODE_LE(x_1, x_2)) = 0 POL(ENCODE_MINUS(x_1, x_2)) = 0 POL(ENCODE_PRED(x_1)) = 0 POL(GCD(x_1, x_2)) = [1] POL(IF(x_1, x_2, x_3)) = [1] POL(LE(x_1, x_2)) = 0 POL(MINUS(x_1, x_2)) = 0 POL(PRED(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19) = 0 POL(c20) = 0 POL(c21(x_1)) = x_1 POL(c22) = 0 POL(c23) = 0 POL(c24) = 0 POL(c25) = 0 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c27(x_1, x_2)) = x_1 + x_2 POL(c28(x_1, x_2)) = x_1 + x_2 POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(cons_gcd(x_1, x_2)) = [1] + x_1 + x_2 POL(cons_if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(cons_le(x_1, x_2)) = [1] + x_1 + x_2 POL(cons_minus(x_1, x_2)) = [1] + x_1 + x_2 POL(cons_pred(x_1)) = [1] + x_1 POL(encArg(x_1)) = [1] + x_1 POL(false) = 0 POL(gcd(x_1, x_2)) = [1] + x_1 + x_2 POL(if(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(minus(x_1, x_2)) = x_2 POL(pred(x_1)) = [1] + x_1 POL(s(x_1)) = x_1 POL(true) = 0 ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) encArg(cons_pred(z0)) -> pred(encArg(z0)) encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) minus(z0, s(z1)) -> pred(minus(z0, z1)) minus(z0, 0) -> z0 pred(s(z0)) -> z0 le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true gcd(0, z0) -> 0 gcd(s(z0), 0) -> s(z0) gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) S tuples: MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) K tuples: GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 Defined Rule Symbols: encArg_1, minus_2, pred_1, le_2, gcd_2, if_3 Defined Pair Symbols: ENCARG_1, MINUS_2, PRED_1, LE_2, GCD_2, IF_3, ENCODE_MINUS_2, ENCODE_PRED_1, ENCODE_LE_2, ENCODE_GCD_2, ENCODE_IF_3 Compound Symbols: c_1, c4_3, c5_2, c6_3, c7_3, c8_4, c18_2, c19, c20, c21_1, c22, c23, c24, c25, c26_2, c27_2, c28_2, c1_1 ---------------------------------------- (17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MINUS(z0, 0) -> c19 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) We considered the (Usable) Rules: gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) le(s(z0), s(z1)) -> le(z0, z1) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) le(s(z0), 0) -> false encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) gcd(s(z0), 0) -> s(z0) encArg(cons_pred(z0)) -> pred(encArg(z0)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) encArg(0) -> 0 encArg(true) -> true encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) pred(s(z0)) -> z0 encArg(s(z0)) -> s(encArg(z0)) encArg(false) -> false minus(z0, s(z1)) -> pred(minus(z0, z1)) gcd(0, z0) -> 0 le(0, z0) -> true minus(z0, 0) -> z0 And the Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ENCARG(x_1)) = x_1 + x_1^2 POL(ENCODE_GCD(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2 POL(ENCODE_IF(x_1, x_2, x_3)) = [2] + [2]x_1 + x_2 + x_3 + [2]x_3^2 + [2]x_2*x_3 + x_1*x_3 + x_1^2 + [2]x_1*x_2 + x_2^2 POL(ENCODE_LE(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + x_2^2 + x_1*x_2 + [2]x_1^2 POL(ENCODE_MINUS(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + [2]x_2^2 + x_1*x_2 + [2]x_1^2 POL(ENCODE_PRED(x_1)) = [1] + x_1 + [2]x_1^2 POL(GCD(x_1, x_2)) = [2] + [2]x_1*x_2 POL(IF(x_1, x_2, x_3)) = [2]x_2*x_3 POL(LE(x_1, x_2)) = 0 POL(MINUS(x_1, x_2)) = [2] POL(PRED(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19) = 0 POL(c20) = 0 POL(c21(x_1)) = x_1 POL(c22) = 0 POL(c23) = 0 POL(c24) = 0 POL(c25) = 0 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c27(x_1, x_2)) = x_1 + x_2 POL(c28(x_1, x_2)) = x_1 + x_2 POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(cons_gcd(x_1, x_2)) = [1] + x_1 + x_2 POL(cons_if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(cons_le(x_1, x_2)) = x_1 + x_2 POL(cons_minus(x_1, x_2)) = [1] + x_1 + x_2 POL(cons_pred(x_1)) = x_1 POL(encArg(x_1)) = x_1 POL(false) = 0 POL(gcd(x_1, x_2)) = x_1 + x_2 POL(if(x_1, x_2, x_3)) = x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(minus(x_1, x_2)) = [1] + x_1 POL(pred(x_1)) = x_1 POL(s(x_1)) = [2] + x_1 POL(true) = 0 ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) encArg(cons_pred(z0)) -> pred(encArg(z0)) encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) minus(z0, s(z1)) -> pred(minus(z0, z1)) minus(z0, 0) -> z0 pred(s(z0)) -> z0 le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true gcd(0, z0) -> 0 gcd(s(z0), 0) -> s(z0) gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) S tuples: MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) K tuples: GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 MINUS(z0, 0) -> c19 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) Defined Rule Symbols: encArg_1, minus_2, pred_1, le_2, gcd_2, if_3 Defined Pair Symbols: ENCARG_1, MINUS_2, PRED_1, LE_2, GCD_2, IF_3, ENCODE_MINUS_2, ENCODE_PRED_1, ENCODE_LE_2, ENCODE_GCD_2, ENCODE_IF_3 Compound Symbols: c_1, c4_3, c5_2, c6_3, c7_3, c8_4, c18_2, c19, c20, c21_1, c22, c23, c24, c25, c26_2, c27_2, c28_2, c1_1 ---------------------------------------- (19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) We considered the (Usable) Rules: gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) le(s(z0), s(z1)) -> le(z0, z1) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) le(s(z0), 0) -> false encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) gcd(s(z0), 0) -> s(z0) encArg(cons_pred(z0)) -> pred(encArg(z0)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) encArg(0) -> 0 encArg(true) -> true encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) pred(s(z0)) -> z0 encArg(s(z0)) -> s(encArg(z0)) encArg(false) -> false minus(z0, s(z1)) -> pred(minus(z0, z1)) gcd(0, z0) -> 0 le(0, z0) -> true minus(z0, 0) -> z0 And the Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ENCARG(x_1)) = x_1^2 POL(ENCODE_GCD(x_1, x_2)) = [2] + x_1 + [2]x_2 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2 POL(ENCODE_IF(x_1, x_2, x_3)) = [1] + [2]x_1 + [2]x_2 + [2]x_3 + [2]x_3^2 + [2]x_2*x_3 + [2]x_1*x_3 + [2]x_1^2 + [2]x_1*x_2 + x_2^2 POL(ENCODE_LE(x_1, x_2)) = [1] + x_1 + [2]x_2 + x_2^2 + x_1*x_2 + x_1^2 POL(ENCODE_MINUS(x_1, x_2)) = [1] + [2]x_1 + x_2 + [2]x_2^2 + x_1*x_2 + [2]x_1^2 POL(ENCODE_PRED(x_1)) = [1] + [2]x_1 + [2]x_1^2 POL(GCD(x_1, x_2)) = [2]x_1*x_2 POL(IF(x_1, x_2, x_3)) = [2]x_2*x_3 POL(LE(x_1, x_2)) = 0 POL(MINUS(x_1, x_2)) = 0 POL(PRED(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19) = 0 POL(c20) = 0 POL(c21(x_1)) = x_1 POL(c22) = 0 POL(c23) = 0 POL(c24) = 0 POL(c25) = 0 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c27(x_1, x_2)) = x_1 + x_2 POL(c28(x_1, x_2)) = x_1 + x_2 POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(cons_gcd(x_1, x_2)) = x_1 + x_2 POL(cons_if(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(cons_le(x_1, x_2)) = [1] + x_1 + x_2 POL(cons_minus(x_1, x_2)) = x_1 + x_2 POL(cons_pred(x_1)) = x_1 POL(encArg(x_1)) = x_1 POL(false) = 0 POL(gcd(x_1, x_2)) = x_1 + x_2 POL(if(x_1, x_2, x_3)) = x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(minus(x_1, x_2)) = x_1 POL(pred(x_1)) = x_1 POL(s(x_1)) = [2] + x_1 POL(true) = 0 ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) encArg(cons_pred(z0)) -> pred(encArg(z0)) encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) minus(z0, s(z1)) -> pred(minus(z0, z1)) minus(z0, 0) -> z0 pred(s(z0)) -> z0 le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true gcd(0, z0) -> 0 gcd(s(z0), 0) -> s(z0) gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) S tuples: MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 K tuples: GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 MINUS(z0, 0) -> c19 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) Defined Rule Symbols: encArg_1, minus_2, pred_1, le_2, gcd_2, if_3 Defined Pair Symbols: ENCARG_1, MINUS_2, PRED_1, LE_2, GCD_2, IF_3, ENCODE_MINUS_2, ENCODE_PRED_1, ENCODE_LE_2, ENCODE_GCD_2, ENCODE_IF_3 Compound Symbols: c_1, c4_3, c5_2, c6_3, c7_3, c8_4, c18_2, c19, c20, c21_1, c22, c23, c24, c25, c26_2, c27_2, c28_2, c1_1 ---------------------------------------- (21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LE(s(z0), 0) -> c22 LE(0, z0) -> c23 We considered the (Usable) Rules: gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) le(s(z0), s(z1)) -> le(z0, z1) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) le(s(z0), 0) -> false encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) gcd(s(z0), 0) -> s(z0) encArg(cons_pred(z0)) -> pred(encArg(z0)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) encArg(0) -> 0 encArg(true) -> true encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) pred(s(z0)) -> z0 encArg(s(z0)) -> s(encArg(z0)) encArg(false) -> false minus(z0, s(z1)) -> pred(minus(z0, z1)) gcd(0, z0) -> 0 le(0, z0) -> true minus(z0, 0) -> z0 And the Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ENCARG(x_1)) = [1] + [2]x_1 + x_1^2 POL(ENCODE_GCD(x_1, x_2)) = [1] + [2]x_1 + x_2 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2 POL(ENCODE_IF(x_1, x_2, x_3)) = [2] + [2]x_1 + [2]x_2 + x_3 + [2]x_3^2 + [2]x_2*x_3 + [2]x_1*x_3 + [2]x_1^2 + [2]x_1*x_2 + [2]x_2^2 POL(ENCODE_LE(x_1, x_2)) = [1] + x_1 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2 POL(ENCODE_MINUS(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2 POL(ENCODE_PRED(x_1)) = [2] + [2]x_1 + [2]x_1^2 POL(GCD(x_1, x_2)) = [1] + [2]x_1*x_2 POL(IF(x_1, x_2, x_3)) = [2]x_2*x_3 POL(LE(x_1, x_2)) = [1] POL(MINUS(x_1, x_2)) = [1] POL(PRED(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19) = 0 POL(c20) = 0 POL(c21(x_1)) = x_1 POL(c22) = 0 POL(c23) = 0 POL(c24) = 0 POL(c25) = 0 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c27(x_1, x_2)) = x_1 + x_2 POL(c28(x_1, x_2)) = x_1 + x_2 POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(cons_gcd(x_1, x_2)) = [1] + x_1 + x_2 POL(cons_if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(cons_le(x_1, x_2)) = [1] + x_1 + x_2 POL(cons_minus(x_1, x_2)) = [2] + x_1 + x_2 POL(cons_pred(x_1)) = x_1 POL(encArg(x_1)) = x_1 POL(false) = 0 POL(gcd(x_1, x_2)) = x_1 + x_2 POL(if(x_1, x_2, x_3)) = x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(minus(x_1, x_2)) = x_1 POL(pred(x_1)) = x_1 POL(s(x_1)) = [2] + x_1 POL(true) = 0 ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) encArg(cons_pred(z0)) -> pred(encArg(z0)) encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) minus(z0, s(z1)) -> pred(minus(z0, z1)) minus(z0, 0) -> z0 pred(s(z0)) -> z0 le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true gcd(0, z0) -> 0 gcd(s(z0), 0) -> s(z0) gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) S tuples: MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) K tuples: GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 MINUS(z0, 0) -> c19 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 Defined Rule Symbols: encArg_1, minus_2, pred_1, le_2, gcd_2, if_3 Defined Pair Symbols: ENCARG_1, MINUS_2, PRED_1, LE_2, GCD_2, IF_3, ENCODE_MINUS_2, ENCODE_PRED_1, ENCODE_LE_2, ENCODE_GCD_2, ENCODE_IF_3 Compound Symbols: c_1, c4_3, c5_2, c6_3, c7_3, c8_4, c18_2, c19, c20, c21_1, c22, c23, c24, c25, c26_2, c27_2, c28_2, c1_1 ---------------------------------------- (23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LE(s(z0), s(z1)) -> c21(LE(z0, z1)) We considered the (Usable) Rules: gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) le(s(z0), s(z1)) -> le(z0, z1) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) le(s(z0), 0) -> false encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) gcd(s(z0), 0) -> s(z0) encArg(cons_pred(z0)) -> pred(encArg(z0)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) encArg(0) -> 0 encArg(true) -> true encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) pred(s(z0)) -> z0 encArg(s(z0)) -> s(encArg(z0)) encArg(false) -> false minus(z0, s(z1)) -> pred(minus(z0, z1)) gcd(0, z0) -> 0 le(0, z0) -> true minus(z0, 0) -> z0 And the Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ENCARG(x_1)) = x_1^2 POL(ENCODE_GCD(x_1, x_2)) = [1] + [2]x_1 + [2]x_2 + x_2^2 + [2]x_1*x_2 + [2]x_1^2 POL(ENCODE_IF(x_1, x_2, x_3)) = [1] + x_1 + x_2 + [2]x_3 + [2]x_3^2 + [2]x_2*x_3 + x_1*x_3 + [2]x_1^2 + x_1*x_2 + [2]x_2^2 POL(ENCODE_LE(x_1, x_2)) = [1] + x_1 + [2]x_2 + [2]x_2^2 + x_1*x_2 + [2]x_1^2 POL(ENCODE_MINUS(x_1, x_2)) = [2] + [2]x_1 + x_2 + x_2^2 + [2]x_1*x_2 + x_1^2 POL(ENCODE_PRED(x_1)) = [1] + [2]x_1 + [2]x_1^2 POL(GCD(x_1, x_2)) = [2]x_2 + [2]x_1*x_2 POL(IF(x_1, x_2, x_3)) = x_3 + [2]x_2*x_3 POL(LE(x_1, x_2)) = x_1 POL(MINUS(x_1, x_2)) = 0 POL(PRED(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19) = 0 POL(c20) = 0 POL(c21(x_1)) = x_1 POL(c22) = 0 POL(c23) = 0 POL(c24) = 0 POL(c25) = 0 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c27(x_1, x_2)) = x_1 + x_2 POL(c28(x_1, x_2)) = x_1 + x_2 POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(cons_gcd(x_1, x_2)) = [1] + x_1 + x_2 POL(cons_if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(cons_le(x_1, x_2)) = [1] + x_1 + x_2 POL(cons_minus(x_1, x_2)) = [1] + x_1 + x_2 POL(cons_pred(x_1)) = x_1 POL(encArg(x_1)) = x_1 POL(false) = 0 POL(gcd(x_1, x_2)) = [1] + x_1 + x_2 POL(if(x_1, x_2, x_3)) = x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(minus(x_1, x_2)) = x_1 POL(pred(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) encArg(cons_pred(z0)) -> pred(encArg(z0)) encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) minus(z0, s(z1)) -> pred(minus(z0, z1)) minus(z0, 0) -> z0 pred(s(z0)) -> z0 le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true gcd(0, z0) -> 0 gcd(s(z0), 0) -> s(z0) gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) S tuples: MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) PRED(s(z0)) -> c20 K tuples: GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 MINUS(z0, 0) -> c19 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) Defined Rule Symbols: encArg_1, minus_2, pred_1, le_2, gcd_2, if_3 Defined Pair Symbols: ENCARG_1, MINUS_2, PRED_1, LE_2, GCD_2, IF_3, ENCODE_MINUS_2, ENCODE_PRED_1, ENCODE_LE_2, ENCODE_GCD_2, ENCODE_IF_3 Compound Symbols: c_1, c4_3, c5_2, c6_3, c7_3, c8_4, c18_2, c19, c20, c21_1, c22, c23, c24, c25, c26_2, c27_2, c28_2, c1_1 ---------------------------------------- (25) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) We considered the (Usable) Rules: gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) le(s(z0), s(z1)) -> le(z0, z1) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) le(s(z0), 0) -> false encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) gcd(s(z0), 0) -> s(z0) encArg(cons_pred(z0)) -> pred(encArg(z0)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) encArg(0) -> 0 encArg(true) -> true encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) pred(s(z0)) -> z0 encArg(s(z0)) -> s(encArg(z0)) encArg(false) -> false minus(z0, s(z1)) -> pred(minus(z0, z1)) gcd(0, z0) -> 0 le(0, z0) -> true minus(z0, 0) -> z0 And the Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ENCARG(x_1)) = x_1^2 POL(ENCODE_GCD(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2 POL(ENCODE_IF(x_1, x_2, x_3)) = [2] + [2]x_1 + [2]x_2 + [2]x_3 + [2]x_3^2 + [2]x_2*x_3 + [2]x_1*x_3 + [2]x_1^2 + [2]x_1*x_2 + [2]x_2^2 POL(ENCODE_LE(x_1, x_2)) = [1] + [2]x_1 + [2]x_2^2 + [2]x_1*x_2 + x_1^2 POL(ENCODE_MINUS(x_1, x_2)) = [1] + [2]x_1 + [2]x_2 + [2]x_2^2 + x_1*x_2 + [2]x_1^2 POL(ENCODE_PRED(x_1)) = [1] + x_1^2 POL(GCD(x_1, x_2)) = x_2 + [2]x_1*x_2 POL(IF(x_1, x_2, x_3)) = [2]x_2*x_3 POL(LE(x_1, x_2)) = 0 POL(MINUS(x_1, x_2)) = [1] + x_2 POL(PRED(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19) = 0 POL(c20) = 0 POL(c21(x_1)) = x_1 POL(c22) = 0 POL(c23) = 0 POL(c24) = 0 POL(c25) = 0 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c27(x_1, x_2)) = x_1 + x_2 POL(c28(x_1, x_2)) = x_1 + x_2 POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(cons_gcd(x_1, x_2)) = [2] + x_1 + x_2 POL(cons_if(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(cons_le(x_1, x_2)) = x_1 + x_2 POL(cons_minus(x_1, x_2)) = [2] + x_1 + x_2 POL(cons_pred(x_1)) = x_1 POL(encArg(x_1)) = x_1 POL(false) = 0 POL(gcd(x_1, x_2)) = x_1 + x_2 POL(if(x_1, x_2, x_3)) = x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(minus(x_1, x_2)) = x_1 POL(pred(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) encArg(cons_pred(z0)) -> pred(encArg(z0)) encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) minus(z0, s(z1)) -> pred(minus(z0, z1)) minus(z0, 0) -> z0 pred(s(z0)) -> z0 le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true gcd(0, z0) -> 0 gcd(s(z0), 0) -> s(z0) gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) S tuples: PRED(s(z0)) -> c20 K tuples: GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 MINUS(z0, 0) -> c19 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) Defined Rule Symbols: encArg_1, minus_2, pred_1, le_2, gcd_2, if_3 Defined Pair Symbols: ENCARG_1, MINUS_2, PRED_1, LE_2, GCD_2, IF_3, ENCODE_MINUS_2, ENCODE_PRED_1, ENCODE_LE_2, ENCODE_GCD_2, ENCODE_IF_3 Compound Symbols: c_1, c4_3, c5_2, c6_3, c7_3, c8_4, c18_2, c19, c20, c21_1, c22, c23, c24, c25, c26_2, c27_2, c28_2, c1_1 ---------------------------------------- (27) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. PRED(s(z0)) -> c20 We considered the (Usable) Rules: gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) le(s(z0), s(z1)) -> le(z0, z1) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) le(s(z0), 0) -> false encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) gcd(s(z0), 0) -> s(z0) encArg(cons_pred(z0)) -> pred(encArg(z0)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) encArg(0) -> 0 encArg(true) -> true encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) pred(s(z0)) -> z0 encArg(s(z0)) -> s(encArg(z0)) encArg(false) -> false minus(z0, s(z1)) -> pred(minus(z0, z1)) gcd(0, z0) -> 0 le(0, z0) -> true minus(z0, 0) -> z0 And the Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ENCARG(x_1)) = x_1^2 POL(ENCODE_GCD(x_1, x_2)) = [1] + x_1 + [2]x_2 + [2]x_2^2 + x_1*x_2 + [2]x_1^2 POL(ENCODE_IF(x_1, x_2, x_3)) = [2] + x_1 + [2]x_2 + [2]x_3 + [2]x_3^2 + x_2*x_3 + [2]x_1*x_3 + [2]x_1^2 + [2]x_1*x_2 + x_2^2 POL(ENCODE_LE(x_1, x_2)) = [2] + [2]x_1 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2 POL(ENCODE_MINUS(x_1, x_2)) = [1] + [2]x_1 + [2]x_2 + x_2^2 + x_1*x_2 + [2]x_1^2 POL(ENCODE_PRED(x_1)) = [1] + [2]x_1^2 POL(GCD(x_1, x_2)) = x_1*x_2 POL(IF(x_1, x_2, x_3)) = x_2*x_3 POL(LE(x_1, x_2)) = 0 POL(MINUS(x_1, x_2)) = x_2 POL(PRED(x_1)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19) = 0 POL(c20) = 0 POL(c21(x_1)) = x_1 POL(c22) = 0 POL(c23) = 0 POL(c24) = 0 POL(c25) = 0 POL(c26(x_1, x_2)) = x_1 + x_2 POL(c27(x_1, x_2)) = x_1 + x_2 POL(c28(x_1, x_2)) = x_1 + x_2 POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c8(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(cons_gcd(x_1, x_2)) = x_1 + x_2 POL(cons_if(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(cons_le(x_1, x_2)) = x_1 + x_2 POL(cons_minus(x_1, x_2)) = [1] + x_1 + x_2 POL(cons_pred(x_1)) = [2] + x_1 POL(encArg(x_1)) = x_1 POL(false) = 0 POL(gcd(x_1, x_2)) = x_1 + x_2 POL(if(x_1, x_2, x_3)) = x_2 + x_3 POL(le(x_1, x_2)) = 0 POL(minus(x_1, x_2)) = x_1 POL(pred(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(z0, z1)) -> minus(encArg(z0), encArg(z1)) encArg(cons_pred(z0)) -> pred(encArg(z0)) encArg(cons_le(z0, z1)) -> le(encArg(z0), encArg(z1)) encArg(cons_gcd(z0, z1)) -> gcd(encArg(z0), encArg(z1)) encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2)) minus(z0, s(z1)) -> pred(minus(z0, z1)) minus(z0, 0) -> z0 pred(s(z0)) -> z0 le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true gcd(0, z0) -> 0 gcd(s(z0), 0) -> s(z0) gcd(s(z0), s(z1)) -> if(le(z1, z0), s(z0), s(z1)) if(true, s(z0), s(z1)) -> gcd(minus(z0, z1), s(z1)) if(false, s(z0), s(z1)) -> gcd(minus(z1, z0), s(z0)) Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_minus(z0, z1)) -> c4(MINUS(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_pred(z0)) -> c5(PRED(encArg(z0)), ENCARG(z0)) ENCARG(cons_le(z0, z1)) -> c6(LE(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_gcd(z0, z1)) -> c7(GCD(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_if(z0, z1, z2)) -> c8(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) MINUS(z0, 0) -> c19 PRED(s(z0)) -> c20 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) ENCODE_MINUS(z0, z1) -> c1(MINUS(encArg(z0), encArg(z1))) ENCODE_PRED(z0) -> c1(PRED(encArg(z0))) ENCODE_LE(z0, z1) -> c1(LE(encArg(z0), encArg(z1))) ENCODE_GCD(z0, z1) -> c1(GCD(encArg(z0), encArg(z1))) ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2))) S tuples:none K tuples: GCD(0, z0) -> c24 GCD(s(z0), 0) -> c25 MINUS(z0, 0) -> c19 GCD(s(z0), s(z1)) -> c26(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) IF(true, s(z0), s(z1)) -> c27(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(false, s(z0), s(z1)) -> c28(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) LE(s(z0), 0) -> c22 LE(0, z0) -> c23 LE(s(z0), s(z1)) -> c21(LE(z0, z1)) MINUS(z0, s(z1)) -> c18(PRED(minus(z0, z1)), MINUS(z0, z1)) PRED(s(z0)) -> c20 Defined Rule Symbols: encArg_1, minus_2, pred_1, le_2, gcd_2, if_3 Defined Pair Symbols: ENCARG_1, MINUS_2, PRED_1, LE_2, GCD_2, IF_3, ENCODE_MINUS_2, ENCODE_PRED_1, ENCODE_LE_2, ENCODE_GCD_2, ENCODE_IF_3 Compound Symbols: c_1, c4_3, c5_2, c6_3, c7_3, c8_4, c18_2, c19, c20, c21_1, c22, c23, c24, c25, c26_2, c27_2, c28_2, c1_1 ---------------------------------------- (29) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (30) BOUNDS(1, 1) ---------------------------------------- (31) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (32) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: minus(X, s(Y)) -> pred(minus(X, Y)) minus(X, 0) -> X pred(s(X)) -> X le(s(X), s(Y)) -> le(X, Y) le(s(X), 0) -> false le(0, Y) -> true gcd(0, Y) -> 0 gcd(s(X), 0) -> s(X) gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_pred(x_1)) -> pred(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_pred(x_1) -> pred(encArg(x_1)) encode_0 -> 0 encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_false -> false encode_true -> true encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (33) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence minus(X, s(Y)) ->^+ pred(minus(X, Y)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [Y / s(Y)]. The result substitution is [ ]. ---------------------------------------- (34) Complex Obligation (BEST) ---------------------------------------- (35) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: minus(X, s(Y)) -> pred(minus(X, Y)) minus(X, 0) -> X pred(s(X)) -> X le(s(X), s(Y)) -> le(X, Y) le(s(X), 0) -> false le(0, Y) -> true gcd(0, Y) -> 0 gcd(s(X), 0) -> s(X) gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_pred(x_1)) -> pred(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_pred(x_1) -> pred(encArg(x_1)) encode_0 -> 0 encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_false -> false encode_true -> true encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (36) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (37) BOUNDS(n^1, INF) ---------------------------------------- (38) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: minus(X, s(Y)) -> pred(minus(X, Y)) minus(X, 0) -> X pred(s(X)) -> X le(s(X), s(Y)) -> le(X, Y) le(s(X), 0) -> false le(0, Y) -> true gcd(0, Y) -> 0 gcd(s(X), 0) -> s(X) gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(false) -> false encArg(true) -> true encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_pred(x_1)) -> pred(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_pred(x_1) -> pred(encArg(x_1)) encode_0 -> 0 encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_false -> false encode_true -> true encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST