WORST_CASE(?,O(n^3)) proof of input_I1YIq36w7R.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3). (0) CpxTRS (1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (2) CdtProblem (3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (4) CdtProblem (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CdtProblem (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CdtProblem (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 105 ms] (10) CdtProblem (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 96 ms] (12) CdtProblem (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 734 ms] (14) CdtProblem (15) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (16) BOUNDS(1, 1) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: minus(0, y) -> 0 minus(s(x), 0) -> s(x) minus(s(x), s(y)) -> minus(x, y) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) if(true, x, y) -> x if(false, x, y) -> y perfectp(0) -> false perfectp(s(x)) -> f(x, s(0), s(x), s(x)) f(0, y, 0, u) -> true f(0, y, s(z), u) -> false f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u) f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (2) Obligation: Complexity Dependency Tuples Problem Rules: minus(0, z0) -> 0 minus(s(z0), 0) -> s(z0) minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 perfectp(0) -> false perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0)) f(0, z0, 0, z1) -> true f(0, z0, s(z1), z2) -> false f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2) f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) Tuples: MINUS(0, z0) -> c MINUS(s(z0), 0) -> c1 MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) LE(0, z0) -> c3 LE(s(z0), 0) -> c4 LE(s(z0), s(z1)) -> c5(LE(z0, z1)) IF(true, z0, z1) -> c6 IF(false, z0, z1) -> c7 PERFECTP(0) -> c8 PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0))) F(0, z0, 0, z1) -> c10 F(0, z0, s(z1), z2) -> c11 F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3)) S tuples: MINUS(0, z0) -> c MINUS(s(z0), 0) -> c1 MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) LE(0, z0) -> c3 LE(s(z0), 0) -> c4 LE(s(z0), s(z1)) -> c5(LE(z0, z1)) IF(true, z0, z1) -> c6 IF(false, z0, z1) -> c7 PERFECTP(0) -> c8 PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0))) F(0, z0, 0, z1) -> c10 F(0, z0, s(z1), z2) -> c11 F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3)) K tuples:none Defined Rule Symbols: minus_2, le_2, if_3, perfectp_1, f_4 Defined Pair Symbols: MINUS_2, LE_2, IF_3, PERFECTP_1, F_4 Compound Symbols: c, c1, c2_1, c3, c4, c5_1, c6, c7, c8, c9_1, c10, c11, c12_2, c13_2, c14_3, c15_2 ---------------------------------------- (3) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0))) Removed 9 trailing nodes: F(0, z0, 0, z1) -> c10 MINUS(s(z0), 0) -> c1 F(0, z0, s(z1), z2) -> c11 IF(false, z0, z1) -> c7 PERFECTP(0) -> c8 IF(true, z0, z1) -> c6 LE(s(z0), 0) -> c4 MINUS(0, z0) -> c LE(0, z0) -> c3 ---------------------------------------- (4) Obligation: Complexity Dependency Tuples Problem Rules: minus(0, z0) -> 0 minus(s(z0), 0) -> s(z0) minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 perfectp(0) -> false perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0)) f(0, z0, 0, z1) -> true f(0, z0, s(z1), z2) -> false f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2) f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) Tuples: MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3)) S tuples: MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3)) K tuples:none Defined Rule Symbols: minus_2, le_2, if_3, perfectp_1, f_4 Defined Pair Symbols: MINUS_2, LE_2, F_4 Compound Symbols: c2_1, c5_1, c12_2, c13_2, c14_3, c15_2 ---------------------------------------- (5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: minus(0, z0) -> 0 minus(s(z0), 0) -> s(z0) minus(s(z0), s(z1)) -> minus(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 perfectp(0) -> false perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0)) f(0, z0, 0, z1) -> true f(0, z0, s(z1), z2) -> false f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2) f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) Tuples: MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) S tuples: MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) K tuples:none Defined Rule Symbols: minus_2, le_2, if_3, perfectp_1, f_4 Defined Pair Symbols: MINUS_2, LE_2, F_4 Compound Symbols: c2_1, c5_1, c12_2, c13_1, c14_2, c15_1 ---------------------------------------- (7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 perfectp(0) -> false perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0)) f(0, z0, 0, z1) -> true f(0, z0, s(z1), z2) -> false f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2) f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)) ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(s(z0), 0) -> s(z0) Tuples: MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) S tuples: MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) K tuples:none Defined Rule Symbols: minus_2 Defined Pair Symbols: MINUS_2, LE_2, F_4 Compound Symbols: c2_1, c5_1, c12_2, c13_1, c14_2, c15_1 ---------------------------------------- (9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) We considered the (Usable) Rules:none And the Tuples: MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(F(x_1, x_2, x_3, x_4)) = [1] + x_1 POL(LE(x_1, x_2)) = [1] + x_1 POL(MINUS(x_1, x_2)) = 0 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(minus(x_1, x_2)) = [1] + x_1 + x_2 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(s(z0), 0) -> s(z0) Tuples: MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) S tuples: MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) K tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) Defined Rule Symbols: minus_2 Defined Pair Symbols: MINUS_2, LE_2, F_4 Compound Symbols: c2_1, c5_1, c12_2, c13_1, c14_2, c15_1 ---------------------------------------- (11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) We considered the (Usable) Rules: minus(s(z0), 0) -> s(z0) minus(s(z0), s(z1)) -> minus(z0, z1) minus(0, z0) -> 0 And the Tuples: MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(F(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_1*x_4 POL(LE(x_1, x_2)) = [2] + x_2 POL(MINUS(x_1, x_2)) = 0 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(minus(x_1, x_2)) = x_1 POL(s(x_1)) = [2] + x_1 ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(s(z0), 0) -> s(z0) Tuples: MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) S tuples: MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) K tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) Defined Rule Symbols: minus_2 Defined Pair Symbols: MINUS_2, LE_2, F_4 Compound Symbols: c2_1, c5_1, c12_2, c13_1, c14_2, c15_1 ---------------------------------------- (13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) We considered the (Usable) Rules: minus(s(z0), 0) -> s(z0) minus(s(z0), s(z1)) -> minus(z0, z1) minus(0, z0) -> 0 And the Tuples: MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(F(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_2*x_4 + x_1^2 + x_1*x_2 + x_1*x_3 + x_2*x_3 + x_1^2*x_4 + x_1*x_4^2 + x_1*x_3*x_4 + x_1*x_3^2 POL(LE(x_1, x_2)) = x_1 + x_2 + x_1*x_2 + x_1^2 POL(MINUS(x_1, x_2)) = x_2 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c15(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(minus(x_1, x_2)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(s(z0), 0) -> s(z0) Tuples: MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) S tuples:none K tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0))) F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1)) F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3)) F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0)) MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1)) Defined Rule Symbols: minus_2 Defined Pair Symbols: MINUS_2, LE_2, F_4 Compound Symbols: c2_1, c5_1, c12_2, c13_1, c14_2, c15_1 ---------------------------------------- (15) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (16) BOUNDS(1, 1)