WORST_CASE(Omega(n^1),O(n^1)) proof of input_7JTQH0enWy.trs # AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (2) CdtProblem (3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (4) CdtProblem (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 3 ms] (6) CdtProblem (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 109 ms] (8) CdtProblem (9) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 21 ms] (12) CdtProblem (13) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CdtProblem (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 20 ms] (16) CdtProblem (17) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CdtProblem (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 15 ms] (20) CdtProblem (21) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CdtProblem (23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 16 ms] (24) CdtProblem (25) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 12 ms] (26) CdtProblem (27) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 13 ms] (30) CdtProblem (31) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 14 ms] (32) CdtProblem (33) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 13 ms] (34) CdtProblem (35) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (36) BOUNDS(1, 1) (37) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 45 ms] (38) CdtProblem (39) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CpxRelTRS (41) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CpxRelTRS (43) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (44) typed CpxTrs (45) OrderProof [LOWER BOUND(ID), 44 ms] (46) typed CpxTrs (47) RewriteLemmaProof [LOWER BOUND(ID), 505 ms] (48) BEST (49) proven lower bound (50) LowerBoundPropagationProof [FINISHED, 0 ms] (51) BOUNDS(n^1, INF) (52) typed CpxTrs (53) RewriteLemmaProof [LOWER BOUND(ID), 177 ms] (54) typed CpxTrs (55) RewriteLemmaProof [LOWER BOUND(ID), 90 ms] (56) typed CpxTrs (57) RewriteLemmaProof [LOWER BOUND(ID), 95 ms] (58) typed CpxTrs (59) RewriteLemmaProof [LOWER BOUND(ID), 137 ms] (60) typed CpxTrs (61) RewriteLemmaProof [LOWER BOUND(ID), 101 ms] (62) typed CpxTrs (63) RewriteLemmaProof [LOWER BOUND(ID), 145 ms] (64) typed CpxTrs (65) RewriteLemmaProof [LOWER BOUND(ID), 87 ms] (66) typed CpxTrs (67) RewriteLemmaProof [LOWER BOUND(ID), 160 ms] (68) typed CpxTrs (69) RewriteLemmaProof [LOWER BOUND(ID), 87 ms] (70) typed CpxTrs (71) RewriteLemmaProof [LOWER BOUND(ID), 369 ms] (72) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: f_0(x) -> a f_1(x) -> g_1(x, x) g_1(s(x), y) -> b(f_0(y), g_1(x, y)) f_2(x) -> g_2(x, x) g_2(s(x), y) -> b(f_1(y), g_2(x, y)) f_3(x) -> g_3(x, x) g_3(s(x), y) -> b(f_2(y), g_3(x, y)) f_4(x) -> g_4(x, x) g_4(s(x), y) -> b(f_3(y), g_4(x, y)) f_5(x) -> g_5(x, x) g_5(s(x), y) -> b(f_4(y), g_5(x, y)) f_6(x) -> g_6(x, x) g_6(s(x), y) -> b(f_5(y), g_6(x, y)) f_7(x) -> g_7(x, x) g_7(s(x), y) -> b(f_6(y), g_7(x, y)) f_8(x) -> g_8(x, x) g_8(s(x), y) -> b(f_7(y), g_8(x, y)) f_9(x) -> g_9(x, x) g_9(s(x), y) -> b(f_8(y), g_9(x, y)) f_10(x) -> g_10(x, x) g_10(s(x), y) -> b(f_9(y), g_10(x, y)) 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: f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Tuples: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) K tuples:none Defined Rule Symbols: f_0_1, f_1_1, g_1_2, f_2_1, g_2_2, f_3_1, g_3_2, f_4_1, g_4_2, f_5_1, g_5_2, f_6_1, g_6_2, f_7_1, g_7_2, f_8_1, g_8_2, f_9_1, g_9_2, f_10_1, g_10_2 Defined Pair Symbols: F_0_1, F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, F_10_1, G_10_2 Compound Symbols: c, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1 ---------------------------------------- (3) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: F_10(z0) -> c28(G_10(z0, z0)) Removed 2 trailing nodes: G_1(s(z0), z1) -> c2(F_0(z1)) F_0(z0) -> c ---------------------------------------- (4) Obligation: Complexity Dependency Tuples Problem Rules: f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) K tuples:none Defined Rule Symbols: f_0_1, f_1_1, g_1_2, f_2_1, g_2_2, f_3_1, g_3_2, f_4_1, g_4_2, f_5_1, g_5_2, f_6_1, g_6_2, f_7_1, g_7_2, f_8_1, g_8_2, f_9_1, g_9_2, f_10_1, g_10_2 Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1 ---------------------------------------- (5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1 ---------------------------------------- (7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) We considered the (Usable) Rules:none And the Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(F_1(x_1)) = 0 POL(F_2(x_1)) = 0 POL(F_3(x_1)) = 0 POL(F_4(x_1)) = 0 POL(F_5(x_1)) = 0 POL(F_6(x_1)) = 0 POL(F_7(x_1)) = 0 POL(F_8(x_1)) = 0 POL(F_9(x_1)) = [1] + x_1 POL(G_1(x_1, x_2)) = 0 POL(G_10(x_1, x_2)) = [1] + x_1 + x_2 POL(G_2(x_1, x_2)) = 0 POL(G_3(x_1, x_2)) = 0 POL(G_4(x_1, x_2)) = 0 POL(G_5(x_1, x_2)) = 0 POL(G_6(x_1, x_2)) = 0 POL(G_7(x_1, x_2)) = 0 POL(G_8(x_1, x_2)) = 0 POL(G_9(x_1, x_2)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) K tuples: F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) Defined Rule Symbols:none Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1 ---------------------------------------- (9) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F_8(z0) -> c22(G_8(z0, z0)) ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) K tuples: F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) Defined Rule Symbols:none Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1 ---------------------------------------- (11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) We considered the (Usable) Rules:none And the Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(F_1(x_1)) = 0 POL(F_2(x_1)) = 0 POL(F_3(x_1)) = 0 POL(F_4(x_1)) = 0 POL(F_5(x_1)) = 0 POL(F_6(x_1)) = x_1 POL(F_7(x_1)) = [1] + x_1 POL(F_8(x_1)) = [1] + x_1 POL(F_9(x_1)) = [1] + x_1 POL(G_1(x_1, x_2)) = 0 POL(G_10(x_1, x_2)) = x_1 + x_2 POL(G_2(x_1, x_2)) = 0 POL(G_3(x_1, x_2)) = 0 POL(G_4(x_1, x_2)) = 0 POL(G_5(x_1, x_2)) = 0 POL(G_6(x_1, x_2)) = x_1 POL(G_7(x_1, x_2)) = x_2 POL(G_8(x_1, x_2)) = [1] + x_2 POL(G_9(x_1, x_2)) = [1] + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) K tuples: F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) Defined Rule Symbols:none Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1 ---------------------------------------- (13) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F_5(z0) -> c13(G_5(z0, z0)) ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) K tuples: F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) F_5(z0) -> c13(G_5(z0, z0)) Defined Rule Symbols:none Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_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. G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) We considered the (Usable) Rules:none And the Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(F_1(x_1)) = 0 POL(F_2(x_1)) = 0 POL(F_3(x_1)) = x_1 POL(F_4(x_1)) = x_1 POL(F_5(x_1)) = [1] + x_1 POL(F_6(x_1)) = [1] + x_1 POL(F_7(x_1)) = [1] + x_1 POL(F_8(x_1)) = [1] + x_1 POL(F_9(x_1)) = [1] + x_1 POL(G_1(x_1, x_2)) = 0 POL(G_10(x_1, x_2)) = [1] + x_2 POL(G_2(x_1, x_2)) = 0 POL(G_3(x_1, x_2)) = x_1 POL(G_4(x_1, x_2)) = x_2 POL(G_5(x_1, x_2)) = x_2 POL(G_6(x_1, x_2)) = [1] + x_2 POL(G_7(x_1, x_2)) = [1] + x_2 POL(G_8(x_1, x_2)) = [1] + x_2 POL(G_9(x_1, x_2)) = [1] + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) K tuples: F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) F_5(z0) -> c13(G_5(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) Defined Rule Symbols:none Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1 ---------------------------------------- (17) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F_2(z0) -> c4(G_2(z0, z0)) ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) K tuples: F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) F_5(z0) -> c13(G_5(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) Defined Rule Symbols:none Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1 ---------------------------------------- (19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. G_5(s(z0), z1) -> c14(F_4(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) We considered the (Usable) Rules:none And the Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(F_1(x_1)) = 0 POL(F_2(x_1)) = 0 POL(F_3(x_1)) = 0 POL(F_4(x_1)) = 0 POL(F_5(x_1)) = [1] POL(F_6(x_1)) = [1] POL(F_7(x_1)) = [1] + x_1 POL(F_8(x_1)) = [1] + x_1 POL(F_9(x_1)) = [1] + x_1 POL(G_1(x_1, x_2)) = 0 POL(G_10(x_1, x_2)) = [1] + x_2 POL(G_2(x_1, x_2)) = 0 POL(G_3(x_1, x_2)) = 0 POL(G_4(x_1, x_2)) = 0 POL(G_5(x_1, x_2)) = [1] POL(G_6(x_1, x_2)) = [1] POL(G_7(x_1, x_2)) = x_1 POL(G_8(x_1, x_2)) = [1] + x_2 POL(G_9(x_1, x_2)) = [1] + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) K tuples: F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) F_5(z0) -> c13(G_5(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) Defined Rule Symbols:none Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1 ---------------------------------------- (21) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F_4(z0) -> c10(G_4(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) K tuples: F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) F_5(z0) -> c13(G_5(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) F_6(z0) -> c16(G_6(z0, z0)) Defined Rule Symbols:none Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1 ---------------------------------------- (23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. G_5(s(z0), z1) -> c15(G_5(z0, z1)) We considered the (Usable) Rules:none And the Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(F_1(x_1)) = 0 POL(F_2(x_1)) = 0 POL(F_3(x_1)) = [1] POL(F_4(x_1)) = [1] POL(F_5(x_1)) = [1] + x_1 POL(F_6(x_1)) = [1] + x_1 POL(F_7(x_1)) = [1] + x_1 POL(F_8(x_1)) = [1] + x_1 POL(F_9(x_1)) = [1] + x_1 POL(G_1(x_1, x_2)) = 0 POL(G_10(x_1, x_2)) = [1] + x_2 POL(G_2(x_1, x_2)) = 0 POL(G_3(x_1, x_2)) = [1] POL(G_4(x_1, x_2)) = [1] POL(G_5(x_1, x_2)) = [1] + x_1 POL(G_6(x_1, x_2)) = [1] + x_2 POL(G_7(x_1, x_2)) = [1] + x_2 POL(G_8(x_1, x_2)) = [1] + x_2 POL(G_9(x_1, x_2)) = [1] + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) K tuples: F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) F_5(z0) -> c13(G_5(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) F_6(z0) -> c16(G_6(z0, z0)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) Defined Rule Symbols:none Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1 ---------------------------------------- (25) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. G_4(s(z0), z1) -> c12(G_4(z0, z1)) We considered the (Usable) Rules:none And the Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(F_1(x_1)) = 0 POL(F_2(x_1)) = [1] POL(F_3(x_1)) = [1] POL(F_4(x_1)) = [1] + x_1 POL(F_5(x_1)) = [1] + x_1 POL(F_6(x_1)) = [1] + x_1 POL(F_7(x_1)) = [1] + x_1 POL(F_8(x_1)) = [1] + x_1 POL(F_9(x_1)) = [1] + x_1 POL(G_1(x_1, x_2)) = 0 POL(G_10(x_1, x_2)) = x_1 + x_2 POL(G_2(x_1, x_2)) = 0 POL(G_3(x_1, x_2)) = [1] POL(G_4(x_1, x_2)) = x_1 POL(G_5(x_1, x_2)) = [1] + x_2 POL(G_6(x_1, x_2)) = [1] + x_2 POL(G_7(x_1, x_2)) = [1] + x_2 POL(G_8(x_1, x_2)) = [1] + x_2 POL(G_9(x_1, x_2)) = [1] + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) K tuples: F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) F_5(z0) -> c13(G_5(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) F_6(z0) -> c16(G_6(z0, z0)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) Defined Rule Symbols:none Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1 ---------------------------------------- (27) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: G_4(s(z0), z1) -> c11(F_3(z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) K tuples: F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) F_5(z0) -> c13(G_5(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) F_6(z0) -> c16(G_6(z0, z0)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) G_4(s(z0), z1) -> c11(F_3(z1)) F_3(z0) -> c7(G_3(z0, z0)) Defined Rule Symbols:none Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1 ---------------------------------------- (29) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) We considered the (Usable) Rules:none And the Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(F_1(x_1)) = [1] + x_1 POL(F_2(x_1)) = [1] + x_1 POL(F_3(x_1)) = [1] + x_1 POL(F_4(x_1)) = [1] + x_1 POL(F_5(x_1)) = [1] + x_1 POL(F_6(x_1)) = [1] + x_1 POL(F_7(x_1)) = [1] + x_1 POL(F_8(x_1)) = [1] + x_1 POL(F_9(x_1)) = [1] + x_1 POL(G_1(x_1, x_2)) = x_1 POL(G_10(x_1, x_2)) = x_1 + x_2 POL(G_2(x_1, x_2)) = [1] + x_2 POL(G_3(x_1, x_2)) = [1] + x_2 POL(G_4(x_1, x_2)) = [1] + x_2 POL(G_5(x_1, x_2)) = [1] + x_2 POL(G_6(x_1, x_2)) = [1] + x_2 POL(G_7(x_1, x_2)) = [1] + x_2 POL(G_8(x_1, x_2)) = [1] + x_2 POL(G_9(x_1, x_2)) = [1] + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) K tuples: F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) F_5(z0) -> c13(G_5(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) F_6(z0) -> c16(G_6(z0, z0)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) G_4(s(z0), z1) -> c11(F_3(z1)) F_3(z0) -> c7(G_3(z0, z0)) F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) Defined Rule Symbols:none Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1 ---------------------------------------- (31) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) We considered the (Usable) Rules:none And the Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(F_1(x_1)) = [1] POL(F_2(x_1)) = [1] + x_1 POL(F_3(x_1)) = [1] + x_1 POL(F_4(x_1)) = [1] + x_1 POL(F_5(x_1)) = [1] + x_1 POL(F_6(x_1)) = [1] + x_1 POL(F_7(x_1)) = [1] + x_1 POL(F_8(x_1)) = [1] + x_1 POL(F_9(x_1)) = [1] + x_1 POL(G_1(x_1, x_2)) = [1] POL(G_10(x_1, x_2)) = x_1 + x_2 POL(G_2(x_1, x_2)) = [1] + x_1 POL(G_3(x_1, x_2)) = [1] + x_2 POL(G_4(x_1, x_2)) = [1] + x_2 POL(G_5(x_1, x_2)) = [1] + x_2 POL(G_6(x_1, x_2)) = [1] + x_2 POL(G_7(x_1, x_2)) = [1] + x_2 POL(G_8(x_1, x_2)) = [1] + x_2 POL(G_9(x_1, x_2)) = [1] + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) K tuples: F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) F_5(z0) -> c13(G_5(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) F_6(z0) -> c16(G_6(z0, z0)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) G_4(s(z0), z1) -> c11(F_3(z1)) F_3(z0) -> c7(G_3(z0, z0)) F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) Defined Rule Symbols:none Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1 ---------------------------------------- (33) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) We considered the (Usable) Rules:none And the Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(F_1(x_1)) = [1] POL(F_2(x_1)) = [1] POL(F_3(x_1)) = [1] POL(F_4(x_1)) = [1] POL(F_5(x_1)) = [1] POL(F_6(x_1)) = [1] POL(F_7(x_1)) = [1] POL(F_8(x_1)) = [1] + x_1 POL(F_9(x_1)) = [1] + x_1 POL(G_1(x_1, x_2)) = [1] POL(G_10(x_1, x_2)) = x_1 + x_2 POL(G_2(x_1, x_2)) = [1] POL(G_3(x_1, x_2)) = [1] POL(G_4(x_1, x_2)) = [1] POL(G_5(x_1, x_2)) = [1] POL(G_6(x_1, x_2)) = [1] POL(G_7(x_1, x_2)) = [1] POL(G_8(x_1, x_2)) = [1] + x_1 POL(G_9(x_1, x_2)) = [1] + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples:none K tuples: F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) F_5(z0) -> c13(G_5(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) F_6(z0) -> c16(G_6(z0, z0)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) G_4(s(z0), z1) -> c11(F_3(z1)) F_3(z0) -> c7(G_3(z0, z0)) F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) Defined Rule Symbols:none Defined Pair Symbols: F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, G_10_2 Compound Symbols: c1_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c29_1, c30_1 ---------------------------------------- (35) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (36) BOUNDS(1, 1) ---------------------------------------- (37) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Tuples: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) S tuples: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) K tuples:none Defined Rule Symbols: f_0_1, f_1_1, g_1_2, f_2_1, g_2_2, f_3_1, g_3_2, f_4_1, g_4_2, f_5_1, g_5_2, f_6_1, g_6_2, f_7_1, g_7_2, f_8_1, g_8_2, f_9_1, g_9_2, f_10_1, g_10_2 Defined Pair Symbols: F_0_1, F_1_1, G_1_2, F_2_1, G_2_2, F_3_1, G_3_2, F_4_1, G_4_2, F_5_1, G_5_2, F_6_1, G_6_2, F_7_1, G_7_2, F_8_1, G_8_2, F_9_1, G_9_2, F_10_1, G_10_2 Compound Symbols: c, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1 ---------------------------------------- (39) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (40) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) The (relative) TRS S consists of the following rules: f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Rewrite Strategy: INNERMOST ---------------------------------------- (41) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (42) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) The (relative) TRS S consists of the following rules: f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Rewrite Strategy: INNERMOST ---------------------------------------- (43) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (44) Obligation: Innermost TRS: Rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Types: F_0 :: s -> c c :: c F_1 :: s -> c1 c1 :: c2:c3 -> c1 G_1 :: s -> s -> c2:c3 s :: s -> s c2 :: c -> c2:c3 c3 :: c2:c3 -> c2:c3 F_2 :: s -> c4 c4 :: c5:c6 -> c4 G_2 :: s -> s -> c5:c6 c5 :: c1 -> c5:c6 c6 :: c5:c6 -> c5:c6 F_3 :: s -> c7 c7 :: c8:c9 -> c7 G_3 :: s -> s -> c8:c9 c8 :: c4 -> c8:c9 c9 :: c8:c9 -> c8:c9 F_4 :: s -> c10 c10 :: c11:c12 -> c10 G_4 :: s -> s -> c11:c12 c11 :: c7 -> c11:c12 c12 :: c11:c12 -> c11:c12 F_5 :: s -> c13 c13 :: c14:c15 -> c13 G_5 :: s -> s -> c14:c15 c14 :: c10 -> c14:c15 c15 :: c14:c15 -> c14:c15 F_6 :: s -> c16 c16 :: c17:c18 -> c16 G_6 :: s -> s -> c17:c18 c17 :: c13 -> c17:c18 c18 :: c17:c18 -> c17:c18 F_7 :: s -> c19 c19 :: c20:c21 -> c19 G_7 :: s -> s -> c20:c21 c20 :: c16 -> c20:c21 c21 :: c20:c21 -> c20:c21 F_8 :: s -> c22 c22 :: c23:c24 -> c22 G_8 :: s -> s -> c23:c24 c23 :: c19 -> c23:c24 c24 :: c23:c24 -> c23:c24 F_9 :: s -> c25 c25 :: c26:c27 -> c25 G_9 :: s -> s -> c26:c27 c26 :: c22 -> c26:c27 c27 :: c26:c27 -> c26:c27 F_10 :: s -> c28 c28 :: c29:c30 -> c28 G_10 :: s -> s -> c29:c30 c29 :: c25 -> c29:c30 c30 :: c29:c30 -> c29:c30 f_0 :: s -> a:b a :: a:b f_1 :: s -> a:b g_1 :: s -> s -> a:b b :: a:b -> a:b -> a:b f_2 :: s -> a:b g_2 :: s -> s -> a:b f_3 :: s -> a:b g_3 :: s -> s -> a:b f_4 :: s -> a:b g_4 :: s -> s -> a:b f_5 :: s -> a:b g_5 :: s -> s -> a:b f_6 :: s -> a:b g_6 :: s -> s -> a:b f_7 :: s -> a:b g_7 :: s -> s -> a:b f_8 :: s -> a:b g_8 :: s -> s -> a:b f_9 :: s -> a:b g_9 :: s -> s -> a:b f_10 :: s -> a:b g_10 :: s -> s -> a:b hole_c1_31 :: c hole_s2_31 :: s hole_c13_31 :: c1 hole_c2:c34_31 :: c2:c3 hole_c45_31 :: c4 hole_c5:c66_31 :: c5:c6 hole_c77_31 :: c7 hole_c8:c98_31 :: c8:c9 hole_c109_31 :: c10 hole_c11:c1210_31 :: c11:c12 hole_c1311_31 :: c13 hole_c14:c1512_31 :: c14:c15 hole_c1613_31 :: c16 hole_c17:c1814_31 :: c17:c18 hole_c1915_31 :: c19 hole_c20:c2116_31 :: c20:c21 hole_c2217_31 :: c22 hole_c23:c2418_31 :: c23:c24 hole_c2519_31 :: c25 hole_c26:c2720_31 :: c26:c27 hole_c2821_31 :: c28 hole_c29:c3022_31 :: c29:c30 hole_a:b23_31 :: a:b gen_s24_31 :: Nat -> s gen_c2:c325_31 :: Nat -> c2:c3 gen_c5:c626_31 :: Nat -> c5:c6 gen_c8:c927_31 :: Nat -> c8:c9 gen_c11:c1228_31 :: Nat -> c11:c12 gen_c14:c1529_31 :: Nat -> c14:c15 gen_c17:c1830_31 :: Nat -> c17:c18 gen_c20:c2131_31 :: Nat -> c20:c21 gen_c23:c2432_31 :: Nat -> c23:c24 gen_c26:c2733_31 :: Nat -> c26:c27 gen_c29:c3034_31 :: Nat -> c29:c30 gen_a:b35_31 :: Nat -> a:b ---------------------------------------- (45) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: G_1, G_2, G_3, G_4, G_5, G_6, G_7, G_8, G_9, G_10, g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 ---------------------------------------- (46) Obligation: Innermost TRS: Rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Types: F_0 :: s -> c c :: c F_1 :: s -> c1 c1 :: c2:c3 -> c1 G_1 :: s -> s -> c2:c3 s :: s -> s c2 :: c -> c2:c3 c3 :: c2:c3 -> c2:c3 F_2 :: s -> c4 c4 :: c5:c6 -> c4 G_2 :: s -> s -> c5:c6 c5 :: c1 -> c5:c6 c6 :: c5:c6 -> c5:c6 F_3 :: s -> c7 c7 :: c8:c9 -> c7 G_3 :: s -> s -> c8:c9 c8 :: c4 -> c8:c9 c9 :: c8:c9 -> c8:c9 F_4 :: s -> c10 c10 :: c11:c12 -> c10 G_4 :: s -> s -> c11:c12 c11 :: c7 -> c11:c12 c12 :: c11:c12 -> c11:c12 F_5 :: s -> c13 c13 :: c14:c15 -> c13 G_5 :: s -> s -> c14:c15 c14 :: c10 -> c14:c15 c15 :: c14:c15 -> c14:c15 F_6 :: s -> c16 c16 :: c17:c18 -> c16 G_6 :: s -> s -> c17:c18 c17 :: c13 -> c17:c18 c18 :: c17:c18 -> c17:c18 F_7 :: s -> c19 c19 :: c20:c21 -> c19 G_7 :: s -> s -> c20:c21 c20 :: c16 -> c20:c21 c21 :: c20:c21 -> c20:c21 F_8 :: s -> c22 c22 :: c23:c24 -> c22 G_8 :: s -> s -> c23:c24 c23 :: c19 -> c23:c24 c24 :: c23:c24 -> c23:c24 F_9 :: s -> c25 c25 :: c26:c27 -> c25 G_9 :: s -> s -> c26:c27 c26 :: c22 -> c26:c27 c27 :: c26:c27 -> c26:c27 F_10 :: s -> c28 c28 :: c29:c30 -> c28 G_10 :: s -> s -> c29:c30 c29 :: c25 -> c29:c30 c30 :: c29:c30 -> c29:c30 f_0 :: s -> a:b a :: a:b f_1 :: s -> a:b g_1 :: s -> s -> a:b b :: a:b -> a:b -> a:b f_2 :: s -> a:b g_2 :: s -> s -> a:b f_3 :: s -> a:b g_3 :: s -> s -> a:b f_4 :: s -> a:b g_4 :: s -> s -> a:b f_5 :: s -> a:b g_5 :: s -> s -> a:b f_6 :: s -> a:b g_6 :: s -> s -> a:b f_7 :: s -> a:b g_7 :: s -> s -> a:b f_8 :: s -> a:b g_8 :: s -> s -> a:b f_9 :: s -> a:b g_9 :: s -> s -> a:b f_10 :: s -> a:b g_10 :: s -> s -> a:b hole_c1_31 :: c hole_s2_31 :: s hole_c13_31 :: c1 hole_c2:c34_31 :: c2:c3 hole_c45_31 :: c4 hole_c5:c66_31 :: c5:c6 hole_c77_31 :: c7 hole_c8:c98_31 :: c8:c9 hole_c109_31 :: c10 hole_c11:c1210_31 :: c11:c12 hole_c1311_31 :: c13 hole_c14:c1512_31 :: c14:c15 hole_c1613_31 :: c16 hole_c17:c1814_31 :: c17:c18 hole_c1915_31 :: c19 hole_c20:c2116_31 :: c20:c21 hole_c2217_31 :: c22 hole_c23:c2418_31 :: c23:c24 hole_c2519_31 :: c25 hole_c26:c2720_31 :: c26:c27 hole_c2821_31 :: c28 hole_c29:c3022_31 :: c29:c30 hole_a:b23_31 :: a:b gen_s24_31 :: Nat -> s gen_c2:c325_31 :: Nat -> c2:c3 gen_c5:c626_31 :: Nat -> c5:c6 gen_c8:c927_31 :: Nat -> c8:c9 gen_c11:c1228_31 :: Nat -> c11:c12 gen_c14:c1529_31 :: Nat -> c14:c15 gen_c17:c1830_31 :: Nat -> c17:c18 gen_c20:c2131_31 :: Nat -> c20:c21 gen_c23:c2432_31 :: Nat -> c23:c24 gen_c26:c2733_31 :: Nat -> c26:c27 gen_c29:c3034_31 :: Nat -> c29:c30 gen_a:b35_31 :: Nat -> a:b Generator Equations: gen_s24_31(0) <=> hole_s2_31 gen_s24_31(+(x, 1)) <=> s(gen_s24_31(x)) gen_c2:c325_31(0) <=> c2(c) gen_c2:c325_31(+(x, 1)) <=> c3(gen_c2:c325_31(x)) gen_c5:c626_31(0) <=> c5(c1(c2(c))) gen_c5:c626_31(+(x, 1)) <=> c6(gen_c5:c626_31(x)) gen_c8:c927_31(0) <=> c8(c4(c5(c1(c2(c))))) gen_c8:c927_31(+(x, 1)) <=> c9(gen_c8:c927_31(x)) gen_c11:c1228_31(0) <=> c11(c7(c8(c4(c5(c1(c2(c))))))) gen_c11:c1228_31(+(x, 1)) <=> c12(gen_c11:c1228_31(x)) gen_c14:c1529_31(0) <=> c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))) gen_c14:c1529_31(+(x, 1)) <=> c15(gen_c14:c1529_31(x)) gen_c17:c1830_31(0) <=> c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))) gen_c17:c1830_31(+(x, 1)) <=> c18(gen_c17:c1830_31(x)) gen_c20:c2131_31(0) <=> c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))) gen_c20:c2131_31(+(x, 1)) <=> c21(gen_c20:c2131_31(x)) gen_c23:c2432_31(0) <=> c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))) gen_c23:c2432_31(+(x, 1)) <=> c24(gen_c23:c2432_31(x)) gen_c26:c2733_31(0) <=> c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))) gen_c26:c2733_31(+(x, 1)) <=> c27(gen_c26:c2733_31(x)) gen_c29:c3034_31(0) <=> c29(c25(c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))))) gen_c29:c3034_31(+(x, 1)) <=> c30(gen_c29:c3034_31(x)) gen_a:b35_31(0) <=> a gen_a:b35_31(+(x, 1)) <=> b(a, gen_a:b35_31(x)) The following defined symbols remain to be analysed: G_1, G_2, G_3, G_4, G_5, G_6, G_7, G_8, G_9, G_10, g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 ---------------------------------------- (47) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: G_1(gen_s24_31(+(1, n37_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n37_31) Induction Base: G_1(gen_s24_31(+(1, 0)), gen_s24_31(b)) Induction Step: G_1(gen_s24_31(+(1, +(n37_31, 1))), gen_s24_31(b)) ->_R^Omega(1) c3(G_1(gen_s24_31(+(1, n37_31)), gen_s24_31(b))) ->_IH c3(*36_31) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (48) Complex Obligation (BEST) ---------------------------------------- (49) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Types: F_0 :: s -> c c :: c F_1 :: s -> c1 c1 :: c2:c3 -> c1 G_1 :: s -> s -> c2:c3 s :: s -> s c2 :: c -> c2:c3 c3 :: c2:c3 -> c2:c3 F_2 :: s -> c4 c4 :: c5:c6 -> c4 G_2 :: s -> s -> c5:c6 c5 :: c1 -> c5:c6 c6 :: c5:c6 -> c5:c6 F_3 :: s -> c7 c7 :: c8:c9 -> c7 G_3 :: s -> s -> c8:c9 c8 :: c4 -> c8:c9 c9 :: c8:c9 -> c8:c9 F_4 :: s -> c10 c10 :: c11:c12 -> c10 G_4 :: s -> s -> c11:c12 c11 :: c7 -> c11:c12 c12 :: c11:c12 -> c11:c12 F_5 :: s -> c13 c13 :: c14:c15 -> c13 G_5 :: s -> s -> c14:c15 c14 :: c10 -> c14:c15 c15 :: c14:c15 -> c14:c15 F_6 :: s -> c16 c16 :: c17:c18 -> c16 G_6 :: s -> s -> c17:c18 c17 :: c13 -> c17:c18 c18 :: c17:c18 -> c17:c18 F_7 :: s -> c19 c19 :: c20:c21 -> c19 G_7 :: s -> s -> c20:c21 c20 :: c16 -> c20:c21 c21 :: c20:c21 -> c20:c21 F_8 :: s -> c22 c22 :: c23:c24 -> c22 G_8 :: s -> s -> c23:c24 c23 :: c19 -> c23:c24 c24 :: c23:c24 -> c23:c24 F_9 :: s -> c25 c25 :: c26:c27 -> c25 G_9 :: s -> s -> c26:c27 c26 :: c22 -> c26:c27 c27 :: c26:c27 -> c26:c27 F_10 :: s -> c28 c28 :: c29:c30 -> c28 G_10 :: s -> s -> c29:c30 c29 :: c25 -> c29:c30 c30 :: c29:c30 -> c29:c30 f_0 :: s -> a:b a :: a:b f_1 :: s -> a:b g_1 :: s -> s -> a:b b :: a:b -> a:b -> a:b f_2 :: s -> a:b g_2 :: s -> s -> a:b f_3 :: s -> a:b g_3 :: s -> s -> a:b f_4 :: s -> a:b g_4 :: s -> s -> a:b f_5 :: s -> a:b g_5 :: s -> s -> a:b f_6 :: s -> a:b g_6 :: s -> s -> a:b f_7 :: s -> a:b g_7 :: s -> s -> a:b f_8 :: s -> a:b g_8 :: s -> s -> a:b f_9 :: s -> a:b g_9 :: s -> s -> a:b f_10 :: s -> a:b g_10 :: s -> s -> a:b hole_c1_31 :: c hole_s2_31 :: s hole_c13_31 :: c1 hole_c2:c34_31 :: c2:c3 hole_c45_31 :: c4 hole_c5:c66_31 :: c5:c6 hole_c77_31 :: c7 hole_c8:c98_31 :: c8:c9 hole_c109_31 :: c10 hole_c11:c1210_31 :: c11:c12 hole_c1311_31 :: c13 hole_c14:c1512_31 :: c14:c15 hole_c1613_31 :: c16 hole_c17:c1814_31 :: c17:c18 hole_c1915_31 :: c19 hole_c20:c2116_31 :: c20:c21 hole_c2217_31 :: c22 hole_c23:c2418_31 :: c23:c24 hole_c2519_31 :: c25 hole_c26:c2720_31 :: c26:c27 hole_c2821_31 :: c28 hole_c29:c3022_31 :: c29:c30 hole_a:b23_31 :: a:b gen_s24_31 :: Nat -> s gen_c2:c325_31 :: Nat -> c2:c3 gen_c5:c626_31 :: Nat -> c5:c6 gen_c8:c927_31 :: Nat -> c8:c9 gen_c11:c1228_31 :: Nat -> c11:c12 gen_c14:c1529_31 :: Nat -> c14:c15 gen_c17:c1830_31 :: Nat -> c17:c18 gen_c20:c2131_31 :: Nat -> c20:c21 gen_c23:c2432_31 :: Nat -> c23:c24 gen_c26:c2733_31 :: Nat -> c26:c27 gen_c29:c3034_31 :: Nat -> c29:c30 gen_a:b35_31 :: Nat -> a:b Generator Equations: gen_s24_31(0) <=> hole_s2_31 gen_s24_31(+(x, 1)) <=> s(gen_s24_31(x)) gen_c2:c325_31(0) <=> c2(c) gen_c2:c325_31(+(x, 1)) <=> c3(gen_c2:c325_31(x)) gen_c5:c626_31(0) <=> c5(c1(c2(c))) gen_c5:c626_31(+(x, 1)) <=> c6(gen_c5:c626_31(x)) gen_c8:c927_31(0) <=> c8(c4(c5(c1(c2(c))))) gen_c8:c927_31(+(x, 1)) <=> c9(gen_c8:c927_31(x)) gen_c11:c1228_31(0) <=> c11(c7(c8(c4(c5(c1(c2(c))))))) gen_c11:c1228_31(+(x, 1)) <=> c12(gen_c11:c1228_31(x)) gen_c14:c1529_31(0) <=> c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))) gen_c14:c1529_31(+(x, 1)) <=> c15(gen_c14:c1529_31(x)) gen_c17:c1830_31(0) <=> c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))) gen_c17:c1830_31(+(x, 1)) <=> c18(gen_c17:c1830_31(x)) gen_c20:c2131_31(0) <=> c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))) gen_c20:c2131_31(+(x, 1)) <=> c21(gen_c20:c2131_31(x)) gen_c23:c2432_31(0) <=> c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))) gen_c23:c2432_31(+(x, 1)) <=> c24(gen_c23:c2432_31(x)) gen_c26:c2733_31(0) <=> c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))) gen_c26:c2733_31(+(x, 1)) <=> c27(gen_c26:c2733_31(x)) gen_c29:c3034_31(0) <=> c29(c25(c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))))) gen_c29:c3034_31(+(x, 1)) <=> c30(gen_c29:c3034_31(x)) gen_a:b35_31(0) <=> a gen_a:b35_31(+(x, 1)) <=> b(a, gen_a:b35_31(x)) The following defined symbols remain to be analysed: G_1, G_2, G_3, G_4, G_5, G_6, G_7, G_8, G_9, G_10, g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 ---------------------------------------- (50) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (51) BOUNDS(n^1, INF) ---------------------------------------- (52) Obligation: Innermost TRS: Rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Types: F_0 :: s -> c c :: c F_1 :: s -> c1 c1 :: c2:c3 -> c1 G_1 :: s -> s -> c2:c3 s :: s -> s c2 :: c -> c2:c3 c3 :: c2:c3 -> c2:c3 F_2 :: s -> c4 c4 :: c5:c6 -> c4 G_2 :: s -> s -> c5:c6 c5 :: c1 -> c5:c6 c6 :: c5:c6 -> c5:c6 F_3 :: s -> c7 c7 :: c8:c9 -> c7 G_3 :: s -> s -> c8:c9 c8 :: c4 -> c8:c9 c9 :: c8:c9 -> c8:c9 F_4 :: s -> c10 c10 :: c11:c12 -> c10 G_4 :: s -> s -> c11:c12 c11 :: c7 -> c11:c12 c12 :: c11:c12 -> c11:c12 F_5 :: s -> c13 c13 :: c14:c15 -> c13 G_5 :: s -> s -> c14:c15 c14 :: c10 -> c14:c15 c15 :: c14:c15 -> c14:c15 F_6 :: s -> c16 c16 :: c17:c18 -> c16 G_6 :: s -> s -> c17:c18 c17 :: c13 -> c17:c18 c18 :: c17:c18 -> c17:c18 F_7 :: s -> c19 c19 :: c20:c21 -> c19 G_7 :: s -> s -> c20:c21 c20 :: c16 -> c20:c21 c21 :: c20:c21 -> c20:c21 F_8 :: s -> c22 c22 :: c23:c24 -> c22 G_8 :: s -> s -> c23:c24 c23 :: c19 -> c23:c24 c24 :: c23:c24 -> c23:c24 F_9 :: s -> c25 c25 :: c26:c27 -> c25 G_9 :: s -> s -> c26:c27 c26 :: c22 -> c26:c27 c27 :: c26:c27 -> c26:c27 F_10 :: s -> c28 c28 :: c29:c30 -> c28 G_10 :: s -> s -> c29:c30 c29 :: c25 -> c29:c30 c30 :: c29:c30 -> c29:c30 f_0 :: s -> a:b a :: a:b f_1 :: s -> a:b g_1 :: s -> s -> a:b b :: a:b -> a:b -> a:b f_2 :: s -> a:b g_2 :: s -> s -> a:b f_3 :: s -> a:b g_3 :: s -> s -> a:b f_4 :: s -> a:b g_4 :: s -> s -> a:b f_5 :: s -> a:b g_5 :: s -> s -> a:b f_6 :: s -> a:b g_6 :: s -> s -> a:b f_7 :: s -> a:b g_7 :: s -> s -> a:b f_8 :: s -> a:b g_8 :: s -> s -> a:b f_9 :: s -> a:b g_9 :: s -> s -> a:b f_10 :: s -> a:b g_10 :: s -> s -> a:b hole_c1_31 :: c hole_s2_31 :: s hole_c13_31 :: c1 hole_c2:c34_31 :: c2:c3 hole_c45_31 :: c4 hole_c5:c66_31 :: c5:c6 hole_c77_31 :: c7 hole_c8:c98_31 :: c8:c9 hole_c109_31 :: c10 hole_c11:c1210_31 :: c11:c12 hole_c1311_31 :: c13 hole_c14:c1512_31 :: c14:c15 hole_c1613_31 :: c16 hole_c17:c1814_31 :: c17:c18 hole_c1915_31 :: c19 hole_c20:c2116_31 :: c20:c21 hole_c2217_31 :: c22 hole_c23:c2418_31 :: c23:c24 hole_c2519_31 :: c25 hole_c26:c2720_31 :: c26:c27 hole_c2821_31 :: c28 hole_c29:c3022_31 :: c29:c30 hole_a:b23_31 :: a:b gen_s24_31 :: Nat -> s gen_c2:c325_31 :: Nat -> c2:c3 gen_c5:c626_31 :: Nat -> c5:c6 gen_c8:c927_31 :: Nat -> c8:c9 gen_c11:c1228_31 :: Nat -> c11:c12 gen_c14:c1529_31 :: Nat -> c14:c15 gen_c17:c1830_31 :: Nat -> c17:c18 gen_c20:c2131_31 :: Nat -> c20:c21 gen_c23:c2432_31 :: Nat -> c23:c24 gen_c26:c2733_31 :: Nat -> c26:c27 gen_c29:c3034_31 :: Nat -> c29:c30 gen_a:b35_31 :: Nat -> a:b Lemmas: G_1(gen_s24_31(+(1, n37_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n37_31) Generator Equations: gen_s24_31(0) <=> hole_s2_31 gen_s24_31(+(x, 1)) <=> s(gen_s24_31(x)) gen_c2:c325_31(0) <=> c2(c) gen_c2:c325_31(+(x, 1)) <=> c3(gen_c2:c325_31(x)) gen_c5:c626_31(0) <=> c5(c1(c2(c))) gen_c5:c626_31(+(x, 1)) <=> c6(gen_c5:c626_31(x)) gen_c8:c927_31(0) <=> c8(c4(c5(c1(c2(c))))) gen_c8:c927_31(+(x, 1)) <=> c9(gen_c8:c927_31(x)) gen_c11:c1228_31(0) <=> c11(c7(c8(c4(c5(c1(c2(c))))))) gen_c11:c1228_31(+(x, 1)) <=> c12(gen_c11:c1228_31(x)) gen_c14:c1529_31(0) <=> c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))) gen_c14:c1529_31(+(x, 1)) <=> c15(gen_c14:c1529_31(x)) gen_c17:c1830_31(0) <=> c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))) gen_c17:c1830_31(+(x, 1)) <=> c18(gen_c17:c1830_31(x)) gen_c20:c2131_31(0) <=> c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))) gen_c20:c2131_31(+(x, 1)) <=> c21(gen_c20:c2131_31(x)) gen_c23:c2432_31(0) <=> c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))) gen_c23:c2432_31(+(x, 1)) <=> c24(gen_c23:c2432_31(x)) gen_c26:c2733_31(0) <=> c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))) gen_c26:c2733_31(+(x, 1)) <=> c27(gen_c26:c2733_31(x)) gen_c29:c3034_31(0) <=> c29(c25(c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))))) gen_c29:c3034_31(+(x, 1)) <=> c30(gen_c29:c3034_31(x)) gen_a:b35_31(0) <=> a gen_a:b35_31(+(x, 1)) <=> b(a, gen_a:b35_31(x)) The following defined symbols remain to be analysed: G_2, G_3, G_4, G_5, G_6, G_7, G_8, G_9, G_10, g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 ---------------------------------------- (53) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: G_2(gen_s24_31(+(1, n1593_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n1593_31) Induction Base: G_2(gen_s24_31(+(1, 0)), gen_s24_31(b)) Induction Step: G_2(gen_s24_31(+(1, +(n1593_31, 1))), gen_s24_31(b)) ->_R^Omega(1) c6(G_2(gen_s24_31(+(1, n1593_31)), gen_s24_31(b))) ->_IH c6(*36_31) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (54) Obligation: Innermost TRS: Rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Types: F_0 :: s -> c c :: c F_1 :: s -> c1 c1 :: c2:c3 -> c1 G_1 :: s -> s -> c2:c3 s :: s -> s c2 :: c -> c2:c3 c3 :: c2:c3 -> c2:c3 F_2 :: s -> c4 c4 :: c5:c6 -> c4 G_2 :: s -> s -> c5:c6 c5 :: c1 -> c5:c6 c6 :: c5:c6 -> c5:c6 F_3 :: s -> c7 c7 :: c8:c9 -> c7 G_3 :: s -> s -> c8:c9 c8 :: c4 -> c8:c9 c9 :: c8:c9 -> c8:c9 F_4 :: s -> c10 c10 :: c11:c12 -> c10 G_4 :: s -> s -> c11:c12 c11 :: c7 -> c11:c12 c12 :: c11:c12 -> c11:c12 F_5 :: s -> c13 c13 :: c14:c15 -> c13 G_5 :: s -> s -> c14:c15 c14 :: c10 -> c14:c15 c15 :: c14:c15 -> c14:c15 F_6 :: s -> c16 c16 :: c17:c18 -> c16 G_6 :: s -> s -> c17:c18 c17 :: c13 -> c17:c18 c18 :: c17:c18 -> c17:c18 F_7 :: s -> c19 c19 :: c20:c21 -> c19 G_7 :: s -> s -> c20:c21 c20 :: c16 -> c20:c21 c21 :: c20:c21 -> c20:c21 F_8 :: s -> c22 c22 :: c23:c24 -> c22 G_8 :: s -> s -> c23:c24 c23 :: c19 -> c23:c24 c24 :: c23:c24 -> c23:c24 F_9 :: s -> c25 c25 :: c26:c27 -> c25 G_9 :: s -> s -> c26:c27 c26 :: c22 -> c26:c27 c27 :: c26:c27 -> c26:c27 F_10 :: s -> c28 c28 :: c29:c30 -> c28 G_10 :: s -> s -> c29:c30 c29 :: c25 -> c29:c30 c30 :: c29:c30 -> c29:c30 f_0 :: s -> a:b a :: a:b f_1 :: s -> a:b g_1 :: s -> s -> a:b b :: a:b -> a:b -> a:b f_2 :: s -> a:b g_2 :: s -> s -> a:b f_3 :: s -> a:b g_3 :: s -> s -> a:b f_4 :: s -> a:b g_4 :: s -> s -> a:b f_5 :: s -> a:b g_5 :: s -> s -> a:b f_6 :: s -> a:b g_6 :: s -> s -> a:b f_7 :: s -> a:b g_7 :: s -> s -> a:b f_8 :: s -> a:b g_8 :: s -> s -> a:b f_9 :: s -> a:b g_9 :: s -> s -> a:b f_10 :: s -> a:b g_10 :: s -> s -> a:b hole_c1_31 :: c hole_s2_31 :: s hole_c13_31 :: c1 hole_c2:c34_31 :: c2:c3 hole_c45_31 :: c4 hole_c5:c66_31 :: c5:c6 hole_c77_31 :: c7 hole_c8:c98_31 :: c8:c9 hole_c109_31 :: c10 hole_c11:c1210_31 :: c11:c12 hole_c1311_31 :: c13 hole_c14:c1512_31 :: c14:c15 hole_c1613_31 :: c16 hole_c17:c1814_31 :: c17:c18 hole_c1915_31 :: c19 hole_c20:c2116_31 :: c20:c21 hole_c2217_31 :: c22 hole_c23:c2418_31 :: c23:c24 hole_c2519_31 :: c25 hole_c26:c2720_31 :: c26:c27 hole_c2821_31 :: c28 hole_c29:c3022_31 :: c29:c30 hole_a:b23_31 :: a:b gen_s24_31 :: Nat -> s gen_c2:c325_31 :: Nat -> c2:c3 gen_c5:c626_31 :: Nat -> c5:c6 gen_c8:c927_31 :: Nat -> c8:c9 gen_c11:c1228_31 :: Nat -> c11:c12 gen_c14:c1529_31 :: Nat -> c14:c15 gen_c17:c1830_31 :: Nat -> c17:c18 gen_c20:c2131_31 :: Nat -> c20:c21 gen_c23:c2432_31 :: Nat -> c23:c24 gen_c26:c2733_31 :: Nat -> c26:c27 gen_c29:c3034_31 :: Nat -> c29:c30 gen_a:b35_31 :: Nat -> a:b Lemmas: G_1(gen_s24_31(+(1, n37_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n37_31) G_2(gen_s24_31(+(1, n1593_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n1593_31) Generator Equations: gen_s24_31(0) <=> hole_s2_31 gen_s24_31(+(x, 1)) <=> s(gen_s24_31(x)) gen_c2:c325_31(0) <=> c2(c) gen_c2:c325_31(+(x, 1)) <=> c3(gen_c2:c325_31(x)) gen_c5:c626_31(0) <=> c5(c1(c2(c))) gen_c5:c626_31(+(x, 1)) <=> c6(gen_c5:c626_31(x)) gen_c8:c927_31(0) <=> c8(c4(c5(c1(c2(c))))) gen_c8:c927_31(+(x, 1)) <=> c9(gen_c8:c927_31(x)) gen_c11:c1228_31(0) <=> c11(c7(c8(c4(c5(c1(c2(c))))))) gen_c11:c1228_31(+(x, 1)) <=> c12(gen_c11:c1228_31(x)) gen_c14:c1529_31(0) <=> c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))) gen_c14:c1529_31(+(x, 1)) <=> c15(gen_c14:c1529_31(x)) gen_c17:c1830_31(0) <=> c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))) gen_c17:c1830_31(+(x, 1)) <=> c18(gen_c17:c1830_31(x)) gen_c20:c2131_31(0) <=> c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))) gen_c20:c2131_31(+(x, 1)) <=> c21(gen_c20:c2131_31(x)) gen_c23:c2432_31(0) <=> c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))) gen_c23:c2432_31(+(x, 1)) <=> c24(gen_c23:c2432_31(x)) gen_c26:c2733_31(0) <=> c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))) gen_c26:c2733_31(+(x, 1)) <=> c27(gen_c26:c2733_31(x)) gen_c29:c3034_31(0) <=> c29(c25(c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))))) gen_c29:c3034_31(+(x, 1)) <=> c30(gen_c29:c3034_31(x)) gen_a:b35_31(0) <=> a gen_a:b35_31(+(x, 1)) <=> b(a, gen_a:b35_31(x)) The following defined symbols remain to be analysed: G_3, G_4, G_5, G_6, G_7, G_8, G_9, G_10, g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 ---------------------------------------- (55) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: G_3(gen_s24_31(+(1, n4005_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n4005_31) Induction Base: G_3(gen_s24_31(+(1, 0)), gen_s24_31(b)) Induction Step: G_3(gen_s24_31(+(1, +(n4005_31, 1))), gen_s24_31(b)) ->_R^Omega(1) c9(G_3(gen_s24_31(+(1, n4005_31)), gen_s24_31(b))) ->_IH c9(*36_31) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (56) Obligation: Innermost TRS: Rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Types: F_0 :: s -> c c :: c F_1 :: s -> c1 c1 :: c2:c3 -> c1 G_1 :: s -> s -> c2:c3 s :: s -> s c2 :: c -> c2:c3 c3 :: c2:c3 -> c2:c3 F_2 :: s -> c4 c4 :: c5:c6 -> c4 G_2 :: s -> s -> c5:c6 c5 :: c1 -> c5:c6 c6 :: c5:c6 -> c5:c6 F_3 :: s -> c7 c7 :: c8:c9 -> c7 G_3 :: s -> s -> c8:c9 c8 :: c4 -> c8:c9 c9 :: c8:c9 -> c8:c9 F_4 :: s -> c10 c10 :: c11:c12 -> c10 G_4 :: s -> s -> c11:c12 c11 :: c7 -> c11:c12 c12 :: c11:c12 -> c11:c12 F_5 :: s -> c13 c13 :: c14:c15 -> c13 G_5 :: s -> s -> c14:c15 c14 :: c10 -> c14:c15 c15 :: c14:c15 -> c14:c15 F_6 :: s -> c16 c16 :: c17:c18 -> c16 G_6 :: s -> s -> c17:c18 c17 :: c13 -> c17:c18 c18 :: c17:c18 -> c17:c18 F_7 :: s -> c19 c19 :: c20:c21 -> c19 G_7 :: s -> s -> c20:c21 c20 :: c16 -> c20:c21 c21 :: c20:c21 -> c20:c21 F_8 :: s -> c22 c22 :: c23:c24 -> c22 G_8 :: s -> s -> c23:c24 c23 :: c19 -> c23:c24 c24 :: c23:c24 -> c23:c24 F_9 :: s -> c25 c25 :: c26:c27 -> c25 G_9 :: s -> s -> c26:c27 c26 :: c22 -> c26:c27 c27 :: c26:c27 -> c26:c27 F_10 :: s -> c28 c28 :: c29:c30 -> c28 G_10 :: s -> s -> c29:c30 c29 :: c25 -> c29:c30 c30 :: c29:c30 -> c29:c30 f_0 :: s -> a:b a :: a:b f_1 :: s -> a:b g_1 :: s -> s -> a:b b :: a:b -> a:b -> a:b f_2 :: s -> a:b g_2 :: s -> s -> a:b f_3 :: s -> a:b g_3 :: s -> s -> a:b f_4 :: s -> a:b g_4 :: s -> s -> a:b f_5 :: s -> a:b g_5 :: s -> s -> a:b f_6 :: s -> a:b g_6 :: s -> s -> a:b f_7 :: s -> a:b g_7 :: s -> s -> a:b f_8 :: s -> a:b g_8 :: s -> s -> a:b f_9 :: s -> a:b g_9 :: s -> s -> a:b f_10 :: s -> a:b g_10 :: s -> s -> a:b hole_c1_31 :: c hole_s2_31 :: s hole_c13_31 :: c1 hole_c2:c34_31 :: c2:c3 hole_c45_31 :: c4 hole_c5:c66_31 :: c5:c6 hole_c77_31 :: c7 hole_c8:c98_31 :: c8:c9 hole_c109_31 :: c10 hole_c11:c1210_31 :: c11:c12 hole_c1311_31 :: c13 hole_c14:c1512_31 :: c14:c15 hole_c1613_31 :: c16 hole_c17:c1814_31 :: c17:c18 hole_c1915_31 :: c19 hole_c20:c2116_31 :: c20:c21 hole_c2217_31 :: c22 hole_c23:c2418_31 :: c23:c24 hole_c2519_31 :: c25 hole_c26:c2720_31 :: c26:c27 hole_c2821_31 :: c28 hole_c29:c3022_31 :: c29:c30 hole_a:b23_31 :: a:b gen_s24_31 :: Nat -> s gen_c2:c325_31 :: Nat -> c2:c3 gen_c5:c626_31 :: Nat -> c5:c6 gen_c8:c927_31 :: Nat -> c8:c9 gen_c11:c1228_31 :: Nat -> c11:c12 gen_c14:c1529_31 :: Nat -> c14:c15 gen_c17:c1830_31 :: Nat -> c17:c18 gen_c20:c2131_31 :: Nat -> c20:c21 gen_c23:c2432_31 :: Nat -> c23:c24 gen_c26:c2733_31 :: Nat -> c26:c27 gen_c29:c3034_31 :: Nat -> c29:c30 gen_a:b35_31 :: Nat -> a:b Lemmas: G_1(gen_s24_31(+(1, n37_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n37_31) G_2(gen_s24_31(+(1, n1593_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n1593_31) G_3(gen_s24_31(+(1, n4005_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n4005_31) Generator Equations: gen_s24_31(0) <=> hole_s2_31 gen_s24_31(+(x, 1)) <=> s(gen_s24_31(x)) gen_c2:c325_31(0) <=> c2(c) gen_c2:c325_31(+(x, 1)) <=> c3(gen_c2:c325_31(x)) gen_c5:c626_31(0) <=> c5(c1(c2(c))) gen_c5:c626_31(+(x, 1)) <=> c6(gen_c5:c626_31(x)) gen_c8:c927_31(0) <=> c8(c4(c5(c1(c2(c))))) gen_c8:c927_31(+(x, 1)) <=> c9(gen_c8:c927_31(x)) gen_c11:c1228_31(0) <=> c11(c7(c8(c4(c5(c1(c2(c))))))) gen_c11:c1228_31(+(x, 1)) <=> c12(gen_c11:c1228_31(x)) gen_c14:c1529_31(0) <=> c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))) gen_c14:c1529_31(+(x, 1)) <=> c15(gen_c14:c1529_31(x)) gen_c17:c1830_31(0) <=> c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))) gen_c17:c1830_31(+(x, 1)) <=> c18(gen_c17:c1830_31(x)) gen_c20:c2131_31(0) <=> c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))) gen_c20:c2131_31(+(x, 1)) <=> c21(gen_c20:c2131_31(x)) gen_c23:c2432_31(0) <=> c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))) gen_c23:c2432_31(+(x, 1)) <=> c24(gen_c23:c2432_31(x)) gen_c26:c2733_31(0) <=> c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))) gen_c26:c2733_31(+(x, 1)) <=> c27(gen_c26:c2733_31(x)) gen_c29:c3034_31(0) <=> c29(c25(c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))))) gen_c29:c3034_31(+(x, 1)) <=> c30(gen_c29:c3034_31(x)) gen_a:b35_31(0) <=> a gen_a:b35_31(+(x, 1)) <=> b(a, gen_a:b35_31(x)) The following defined symbols remain to be analysed: G_4, G_5, G_6, G_7, G_8, G_9, G_10, g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 ---------------------------------------- (57) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: G_4(gen_s24_31(+(1, n6737_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n6737_31) Induction Base: G_4(gen_s24_31(+(1, 0)), gen_s24_31(b)) Induction Step: G_4(gen_s24_31(+(1, +(n6737_31, 1))), gen_s24_31(b)) ->_R^Omega(1) c12(G_4(gen_s24_31(+(1, n6737_31)), gen_s24_31(b))) ->_IH c12(*36_31) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (58) Obligation: Innermost TRS: Rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Types: F_0 :: s -> c c :: c F_1 :: s -> c1 c1 :: c2:c3 -> c1 G_1 :: s -> s -> c2:c3 s :: s -> s c2 :: c -> c2:c3 c3 :: c2:c3 -> c2:c3 F_2 :: s -> c4 c4 :: c5:c6 -> c4 G_2 :: s -> s -> c5:c6 c5 :: c1 -> c5:c6 c6 :: c5:c6 -> c5:c6 F_3 :: s -> c7 c7 :: c8:c9 -> c7 G_3 :: s -> s -> c8:c9 c8 :: c4 -> c8:c9 c9 :: c8:c9 -> c8:c9 F_4 :: s -> c10 c10 :: c11:c12 -> c10 G_4 :: s -> s -> c11:c12 c11 :: c7 -> c11:c12 c12 :: c11:c12 -> c11:c12 F_5 :: s -> c13 c13 :: c14:c15 -> c13 G_5 :: s -> s -> c14:c15 c14 :: c10 -> c14:c15 c15 :: c14:c15 -> c14:c15 F_6 :: s -> c16 c16 :: c17:c18 -> c16 G_6 :: s -> s -> c17:c18 c17 :: c13 -> c17:c18 c18 :: c17:c18 -> c17:c18 F_7 :: s -> c19 c19 :: c20:c21 -> c19 G_7 :: s -> s -> c20:c21 c20 :: c16 -> c20:c21 c21 :: c20:c21 -> c20:c21 F_8 :: s -> c22 c22 :: c23:c24 -> c22 G_8 :: s -> s -> c23:c24 c23 :: c19 -> c23:c24 c24 :: c23:c24 -> c23:c24 F_9 :: s -> c25 c25 :: c26:c27 -> c25 G_9 :: s -> s -> c26:c27 c26 :: c22 -> c26:c27 c27 :: c26:c27 -> c26:c27 F_10 :: s -> c28 c28 :: c29:c30 -> c28 G_10 :: s -> s -> c29:c30 c29 :: c25 -> c29:c30 c30 :: c29:c30 -> c29:c30 f_0 :: s -> a:b a :: a:b f_1 :: s -> a:b g_1 :: s -> s -> a:b b :: a:b -> a:b -> a:b f_2 :: s -> a:b g_2 :: s -> s -> a:b f_3 :: s -> a:b g_3 :: s -> s -> a:b f_4 :: s -> a:b g_4 :: s -> s -> a:b f_5 :: s -> a:b g_5 :: s -> s -> a:b f_6 :: s -> a:b g_6 :: s -> s -> a:b f_7 :: s -> a:b g_7 :: s -> s -> a:b f_8 :: s -> a:b g_8 :: s -> s -> a:b f_9 :: s -> a:b g_9 :: s -> s -> a:b f_10 :: s -> a:b g_10 :: s -> s -> a:b hole_c1_31 :: c hole_s2_31 :: s hole_c13_31 :: c1 hole_c2:c34_31 :: c2:c3 hole_c45_31 :: c4 hole_c5:c66_31 :: c5:c6 hole_c77_31 :: c7 hole_c8:c98_31 :: c8:c9 hole_c109_31 :: c10 hole_c11:c1210_31 :: c11:c12 hole_c1311_31 :: c13 hole_c14:c1512_31 :: c14:c15 hole_c1613_31 :: c16 hole_c17:c1814_31 :: c17:c18 hole_c1915_31 :: c19 hole_c20:c2116_31 :: c20:c21 hole_c2217_31 :: c22 hole_c23:c2418_31 :: c23:c24 hole_c2519_31 :: c25 hole_c26:c2720_31 :: c26:c27 hole_c2821_31 :: c28 hole_c29:c3022_31 :: c29:c30 hole_a:b23_31 :: a:b gen_s24_31 :: Nat -> s gen_c2:c325_31 :: Nat -> c2:c3 gen_c5:c626_31 :: Nat -> c5:c6 gen_c8:c927_31 :: Nat -> c8:c9 gen_c11:c1228_31 :: Nat -> c11:c12 gen_c14:c1529_31 :: Nat -> c14:c15 gen_c17:c1830_31 :: Nat -> c17:c18 gen_c20:c2131_31 :: Nat -> c20:c21 gen_c23:c2432_31 :: Nat -> c23:c24 gen_c26:c2733_31 :: Nat -> c26:c27 gen_c29:c3034_31 :: Nat -> c29:c30 gen_a:b35_31 :: Nat -> a:b Lemmas: G_1(gen_s24_31(+(1, n37_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n37_31) G_2(gen_s24_31(+(1, n1593_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n1593_31) G_3(gen_s24_31(+(1, n4005_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n4005_31) G_4(gen_s24_31(+(1, n6737_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n6737_31) Generator Equations: gen_s24_31(0) <=> hole_s2_31 gen_s24_31(+(x, 1)) <=> s(gen_s24_31(x)) gen_c2:c325_31(0) <=> c2(c) gen_c2:c325_31(+(x, 1)) <=> c3(gen_c2:c325_31(x)) gen_c5:c626_31(0) <=> c5(c1(c2(c))) gen_c5:c626_31(+(x, 1)) <=> c6(gen_c5:c626_31(x)) gen_c8:c927_31(0) <=> c8(c4(c5(c1(c2(c))))) gen_c8:c927_31(+(x, 1)) <=> c9(gen_c8:c927_31(x)) gen_c11:c1228_31(0) <=> c11(c7(c8(c4(c5(c1(c2(c))))))) gen_c11:c1228_31(+(x, 1)) <=> c12(gen_c11:c1228_31(x)) gen_c14:c1529_31(0) <=> c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))) gen_c14:c1529_31(+(x, 1)) <=> c15(gen_c14:c1529_31(x)) gen_c17:c1830_31(0) <=> c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))) gen_c17:c1830_31(+(x, 1)) <=> c18(gen_c17:c1830_31(x)) gen_c20:c2131_31(0) <=> c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))) gen_c20:c2131_31(+(x, 1)) <=> c21(gen_c20:c2131_31(x)) gen_c23:c2432_31(0) <=> c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))) gen_c23:c2432_31(+(x, 1)) <=> c24(gen_c23:c2432_31(x)) gen_c26:c2733_31(0) <=> c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))) gen_c26:c2733_31(+(x, 1)) <=> c27(gen_c26:c2733_31(x)) gen_c29:c3034_31(0) <=> c29(c25(c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))))) gen_c29:c3034_31(+(x, 1)) <=> c30(gen_c29:c3034_31(x)) gen_a:b35_31(0) <=> a gen_a:b35_31(+(x, 1)) <=> b(a, gen_a:b35_31(x)) The following defined symbols remain to be analysed: G_5, G_6, G_7, G_8, G_9, G_10, g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 ---------------------------------------- (59) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: G_5(gen_s24_31(+(1, n9789_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n9789_31) Induction Base: G_5(gen_s24_31(+(1, 0)), gen_s24_31(b)) Induction Step: G_5(gen_s24_31(+(1, +(n9789_31, 1))), gen_s24_31(b)) ->_R^Omega(1) c15(G_5(gen_s24_31(+(1, n9789_31)), gen_s24_31(b))) ->_IH c15(*36_31) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (60) Obligation: Innermost TRS: Rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Types: F_0 :: s -> c c :: c F_1 :: s -> c1 c1 :: c2:c3 -> c1 G_1 :: s -> s -> c2:c3 s :: s -> s c2 :: c -> c2:c3 c3 :: c2:c3 -> c2:c3 F_2 :: s -> c4 c4 :: c5:c6 -> c4 G_2 :: s -> s -> c5:c6 c5 :: c1 -> c5:c6 c6 :: c5:c6 -> c5:c6 F_3 :: s -> c7 c7 :: c8:c9 -> c7 G_3 :: s -> s -> c8:c9 c8 :: c4 -> c8:c9 c9 :: c8:c9 -> c8:c9 F_4 :: s -> c10 c10 :: c11:c12 -> c10 G_4 :: s -> s -> c11:c12 c11 :: c7 -> c11:c12 c12 :: c11:c12 -> c11:c12 F_5 :: s -> c13 c13 :: c14:c15 -> c13 G_5 :: s -> s -> c14:c15 c14 :: c10 -> c14:c15 c15 :: c14:c15 -> c14:c15 F_6 :: s -> c16 c16 :: c17:c18 -> c16 G_6 :: s -> s -> c17:c18 c17 :: c13 -> c17:c18 c18 :: c17:c18 -> c17:c18 F_7 :: s -> c19 c19 :: c20:c21 -> c19 G_7 :: s -> s -> c20:c21 c20 :: c16 -> c20:c21 c21 :: c20:c21 -> c20:c21 F_8 :: s -> c22 c22 :: c23:c24 -> c22 G_8 :: s -> s -> c23:c24 c23 :: c19 -> c23:c24 c24 :: c23:c24 -> c23:c24 F_9 :: s -> c25 c25 :: c26:c27 -> c25 G_9 :: s -> s -> c26:c27 c26 :: c22 -> c26:c27 c27 :: c26:c27 -> c26:c27 F_10 :: s -> c28 c28 :: c29:c30 -> c28 G_10 :: s -> s -> c29:c30 c29 :: c25 -> c29:c30 c30 :: c29:c30 -> c29:c30 f_0 :: s -> a:b a :: a:b f_1 :: s -> a:b g_1 :: s -> s -> a:b b :: a:b -> a:b -> a:b f_2 :: s -> a:b g_2 :: s -> s -> a:b f_3 :: s -> a:b g_3 :: s -> s -> a:b f_4 :: s -> a:b g_4 :: s -> s -> a:b f_5 :: s -> a:b g_5 :: s -> s -> a:b f_6 :: s -> a:b g_6 :: s -> s -> a:b f_7 :: s -> a:b g_7 :: s -> s -> a:b f_8 :: s -> a:b g_8 :: s -> s -> a:b f_9 :: s -> a:b g_9 :: s -> s -> a:b f_10 :: s -> a:b g_10 :: s -> s -> a:b hole_c1_31 :: c hole_s2_31 :: s hole_c13_31 :: c1 hole_c2:c34_31 :: c2:c3 hole_c45_31 :: c4 hole_c5:c66_31 :: c5:c6 hole_c77_31 :: c7 hole_c8:c98_31 :: c8:c9 hole_c109_31 :: c10 hole_c11:c1210_31 :: c11:c12 hole_c1311_31 :: c13 hole_c14:c1512_31 :: c14:c15 hole_c1613_31 :: c16 hole_c17:c1814_31 :: c17:c18 hole_c1915_31 :: c19 hole_c20:c2116_31 :: c20:c21 hole_c2217_31 :: c22 hole_c23:c2418_31 :: c23:c24 hole_c2519_31 :: c25 hole_c26:c2720_31 :: c26:c27 hole_c2821_31 :: c28 hole_c29:c3022_31 :: c29:c30 hole_a:b23_31 :: a:b gen_s24_31 :: Nat -> s gen_c2:c325_31 :: Nat -> c2:c3 gen_c5:c626_31 :: Nat -> c5:c6 gen_c8:c927_31 :: Nat -> c8:c9 gen_c11:c1228_31 :: Nat -> c11:c12 gen_c14:c1529_31 :: Nat -> c14:c15 gen_c17:c1830_31 :: Nat -> c17:c18 gen_c20:c2131_31 :: Nat -> c20:c21 gen_c23:c2432_31 :: Nat -> c23:c24 gen_c26:c2733_31 :: Nat -> c26:c27 gen_c29:c3034_31 :: Nat -> c29:c30 gen_a:b35_31 :: Nat -> a:b Lemmas: G_1(gen_s24_31(+(1, n37_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n37_31) G_2(gen_s24_31(+(1, n1593_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n1593_31) G_3(gen_s24_31(+(1, n4005_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n4005_31) G_4(gen_s24_31(+(1, n6737_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n6737_31) G_5(gen_s24_31(+(1, n9789_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n9789_31) Generator Equations: gen_s24_31(0) <=> hole_s2_31 gen_s24_31(+(x, 1)) <=> s(gen_s24_31(x)) gen_c2:c325_31(0) <=> c2(c) gen_c2:c325_31(+(x, 1)) <=> c3(gen_c2:c325_31(x)) gen_c5:c626_31(0) <=> c5(c1(c2(c))) gen_c5:c626_31(+(x, 1)) <=> c6(gen_c5:c626_31(x)) gen_c8:c927_31(0) <=> c8(c4(c5(c1(c2(c))))) gen_c8:c927_31(+(x, 1)) <=> c9(gen_c8:c927_31(x)) gen_c11:c1228_31(0) <=> c11(c7(c8(c4(c5(c1(c2(c))))))) gen_c11:c1228_31(+(x, 1)) <=> c12(gen_c11:c1228_31(x)) gen_c14:c1529_31(0) <=> c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))) gen_c14:c1529_31(+(x, 1)) <=> c15(gen_c14:c1529_31(x)) gen_c17:c1830_31(0) <=> c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))) gen_c17:c1830_31(+(x, 1)) <=> c18(gen_c17:c1830_31(x)) gen_c20:c2131_31(0) <=> c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))) gen_c20:c2131_31(+(x, 1)) <=> c21(gen_c20:c2131_31(x)) gen_c23:c2432_31(0) <=> c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))) gen_c23:c2432_31(+(x, 1)) <=> c24(gen_c23:c2432_31(x)) gen_c26:c2733_31(0) <=> c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))) gen_c26:c2733_31(+(x, 1)) <=> c27(gen_c26:c2733_31(x)) gen_c29:c3034_31(0) <=> c29(c25(c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))))) gen_c29:c3034_31(+(x, 1)) <=> c30(gen_c29:c3034_31(x)) gen_a:b35_31(0) <=> a gen_a:b35_31(+(x, 1)) <=> b(a, gen_a:b35_31(x)) The following defined symbols remain to be analysed: G_6, G_7, G_8, G_9, G_10, g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 ---------------------------------------- (61) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: G_6(gen_s24_31(+(1, n13161_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n13161_31) Induction Base: G_6(gen_s24_31(+(1, 0)), gen_s24_31(b)) Induction Step: G_6(gen_s24_31(+(1, +(n13161_31, 1))), gen_s24_31(b)) ->_R^Omega(1) c18(G_6(gen_s24_31(+(1, n13161_31)), gen_s24_31(b))) ->_IH c18(*36_31) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (62) Obligation: Innermost TRS: Rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Types: F_0 :: s -> c c :: c F_1 :: s -> c1 c1 :: c2:c3 -> c1 G_1 :: s -> s -> c2:c3 s :: s -> s c2 :: c -> c2:c3 c3 :: c2:c3 -> c2:c3 F_2 :: s -> c4 c4 :: c5:c6 -> c4 G_2 :: s -> s -> c5:c6 c5 :: c1 -> c5:c6 c6 :: c5:c6 -> c5:c6 F_3 :: s -> c7 c7 :: c8:c9 -> c7 G_3 :: s -> s -> c8:c9 c8 :: c4 -> c8:c9 c9 :: c8:c9 -> c8:c9 F_4 :: s -> c10 c10 :: c11:c12 -> c10 G_4 :: s -> s -> c11:c12 c11 :: c7 -> c11:c12 c12 :: c11:c12 -> c11:c12 F_5 :: s -> c13 c13 :: c14:c15 -> c13 G_5 :: s -> s -> c14:c15 c14 :: c10 -> c14:c15 c15 :: c14:c15 -> c14:c15 F_6 :: s -> c16 c16 :: c17:c18 -> c16 G_6 :: s -> s -> c17:c18 c17 :: c13 -> c17:c18 c18 :: c17:c18 -> c17:c18 F_7 :: s -> c19 c19 :: c20:c21 -> c19 G_7 :: s -> s -> c20:c21 c20 :: c16 -> c20:c21 c21 :: c20:c21 -> c20:c21 F_8 :: s -> c22 c22 :: c23:c24 -> c22 G_8 :: s -> s -> c23:c24 c23 :: c19 -> c23:c24 c24 :: c23:c24 -> c23:c24 F_9 :: s -> c25 c25 :: c26:c27 -> c25 G_9 :: s -> s -> c26:c27 c26 :: c22 -> c26:c27 c27 :: c26:c27 -> c26:c27 F_10 :: s -> c28 c28 :: c29:c30 -> c28 G_10 :: s -> s -> c29:c30 c29 :: c25 -> c29:c30 c30 :: c29:c30 -> c29:c30 f_0 :: s -> a:b a :: a:b f_1 :: s -> a:b g_1 :: s -> s -> a:b b :: a:b -> a:b -> a:b f_2 :: s -> a:b g_2 :: s -> s -> a:b f_3 :: s -> a:b g_3 :: s -> s -> a:b f_4 :: s -> a:b g_4 :: s -> s -> a:b f_5 :: s -> a:b g_5 :: s -> s -> a:b f_6 :: s -> a:b g_6 :: s -> s -> a:b f_7 :: s -> a:b g_7 :: s -> s -> a:b f_8 :: s -> a:b g_8 :: s -> s -> a:b f_9 :: s -> a:b g_9 :: s -> s -> a:b f_10 :: s -> a:b g_10 :: s -> s -> a:b hole_c1_31 :: c hole_s2_31 :: s hole_c13_31 :: c1 hole_c2:c34_31 :: c2:c3 hole_c45_31 :: c4 hole_c5:c66_31 :: c5:c6 hole_c77_31 :: c7 hole_c8:c98_31 :: c8:c9 hole_c109_31 :: c10 hole_c11:c1210_31 :: c11:c12 hole_c1311_31 :: c13 hole_c14:c1512_31 :: c14:c15 hole_c1613_31 :: c16 hole_c17:c1814_31 :: c17:c18 hole_c1915_31 :: c19 hole_c20:c2116_31 :: c20:c21 hole_c2217_31 :: c22 hole_c23:c2418_31 :: c23:c24 hole_c2519_31 :: c25 hole_c26:c2720_31 :: c26:c27 hole_c2821_31 :: c28 hole_c29:c3022_31 :: c29:c30 hole_a:b23_31 :: a:b gen_s24_31 :: Nat -> s gen_c2:c325_31 :: Nat -> c2:c3 gen_c5:c626_31 :: Nat -> c5:c6 gen_c8:c927_31 :: Nat -> c8:c9 gen_c11:c1228_31 :: Nat -> c11:c12 gen_c14:c1529_31 :: Nat -> c14:c15 gen_c17:c1830_31 :: Nat -> c17:c18 gen_c20:c2131_31 :: Nat -> c20:c21 gen_c23:c2432_31 :: Nat -> c23:c24 gen_c26:c2733_31 :: Nat -> c26:c27 gen_c29:c3034_31 :: Nat -> c29:c30 gen_a:b35_31 :: Nat -> a:b Lemmas: G_1(gen_s24_31(+(1, n37_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n37_31) G_2(gen_s24_31(+(1, n1593_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n1593_31) G_3(gen_s24_31(+(1, n4005_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n4005_31) G_4(gen_s24_31(+(1, n6737_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n6737_31) G_5(gen_s24_31(+(1, n9789_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n9789_31) G_6(gen_s24_31(+(1, n13161_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n13161_31) Generator Equations: gen_s24_31(0) <=> hole_s2_31 gen_s24_31(+(x, 1)) <=> s(gen_s24_31(x)) gen_c2:c325_31(0) <=> c2(c) gen_c2:c325_31(+(x, 1)) <=> c3(gen_c2:c325_31(x)) gen_c5:c626_31(0) <=> c5(c1(c2(c))) gen_c5:c626_31(+(x, 1)) <=> c6(gen_c5:c626_31(x)) gen_c8:c927_31(0) <=> c8(c4(c5(c1(c2(c))))) gen_c8:c927_31(+(x, 1)) <=> c9(gen_c8:c927_31(x)) gen_c11:c1228_31(0) <=> c11(c7(c8(c4(c5(c1(c2(c))))))) gen_c11:c1228_31(+(x, 1)) <=> c12(gen_c11:c1228_31(x)) gen_c14:c1529_31(0) <=> c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))) gen_c14:c1529_31(+(x, 1)) <=> c15(gen_c14:c1529_31(x)) gen_c17:c1830_31(0) <=> c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))) gen_c17:c1830_31(+(x, 1)) <=> c18(gen_c17:c1830_31(x)) gen_c20:c2131_31(0) <=> c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))) gen_c20:c2131_31(+(x, 1)) <=> c21(gen_c20:c2131_31(x)) gen_c23:c2432_31(0) <=> c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))) gen_c23:c2432_31(+(x, 1)) <=> c24(gen_c23:c2432_31(x)) gen_c26:c2733_31(0) <=> c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))) gen_c26:c2733_31(+(x, 1)) <=> c27(gen_c26:c2733_31(x)) gen_c29:c3034_31(0) <=> c29(c25(c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))))) gen_c29:c3034_31(+(x, 1)) <=> c30(gen_c29:c3034_31(x)) gen_a:b35_31(0) <=> a gen_a:b35_31(+(x, 1)) <=> b(a, gen_a:b35_31(x)) The following defined symbols remain to be analysed: G_7, G_8, G_9, G_10, g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 ---------------------------------------- (63) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: G_7(gen_s24_31(+(1, n16853_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n16853_31) Induction Base: G_7(gen_s24_31(+(1, 0)), gen_s24_31(b)) Induction Step: G_7(gen_s24_31(+(1, +(n16853_31, 1))), gen_s24_31(b)) ->_R^Omega(1) c21(G_7(gen_s24_31(+(1, n16853_31)), gen_s24_31(b))) ->_IH c21(*36_31) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (64) Obligation: Innermost TRS: Rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Types: F_0 :: s -> c c :: c F_1 :: s -> c1 c1 :: c2:c3 -> c1 G_1 :: s -> s -> c2:c3 s :: s -> s c2 :: c -> c2:c3 c3 :: c2:c3 -> c2:c3 F_2 :: s -> c4 c4 :: c5:c6 -> c4 G_2 :: s -> s -> c5:c6 c5 :: c1 -> c5:c6 c6 :: c5:c6 -> c5:c6 F_3 :: s -> c7 c7 :: c8:c9 -> c7 G_3 :: s -> s -> c8:c9 c8 :: c4 -> c8:c9 c9 :: c8:c9 -> c8:c9 F_4 :: s -> c10 c10 :: c11:c12 -> c10 G_4 :: s -> s -> c11:c12 c11 :: c7 -> c11:c12 c12 :: c11:c12 -> c11:c12 F_5 :: s -> c13 c13 :: c14:c15 -> c13 G_5 :: s -> s -> c14:c15 c14 :: c10 -> c14:c15 c15 :: c14:c15 -> c14:c15 F_6 :: s -> c16 c16 :: c17:c18 -> c16 G_6 :: s -> s -> c17:c18 c17 :: c13 -> c17:c18 c18 :: c17:c18 -> c17:c18 F_7 :: s -> c19 c19 :: c20:c21 -> c19 G_7 :: s -> s -> c20:c21 c20 :: c16 -> c20:c21 c21 :: c20:c21 -> c20:c21 F_8 :: s -> c22 c22 :: c23:c24 -> c22 G_8 :: s -> s -> c23:c24 c23 :: c19 -> c23:c24 c24 :: c23:c24 -> c23:c24 F_9 :: s -> c25 c25 :: c26:c27 -> c25 G_9 :: s -> s -> c26:c27 c26 :: c22 -> c26:c27 c27 :: c26:c27 -> c26:c27 F_10 :: s -> c28 c28 :: c29:c30 -> c28 G_10 :: s -> s -> c29:c30 c29 :: c25 -> c29:c30 c30 :: c29:c30 -> c29:c30 f_0 :: s -> a:b a :: a:b f_1 :: s -> a:b g_1 :: s -> s -> a:b b :: a:b -> a:b -> a:b f_2 :: s -> a:b g_2 :: s -> s -> a:b f_3 :: s -> a:b g_3 :: s -> s -> a:b f_4 :: s -> a:b g_4 :: s -> s -> a:b f_5 :: s -> a:b g_5 :: s -> s -> a:b f_6 :: s -> a:b g_6 :: s -> s -> a:b f_7 :: s -> a:b g_7 :: s -> s -> a:b f_8 :: s -> a:b g_8 :: s -> s -> a:b f_9 :: s -> a:b g_9 :: s -> s -> a:b f_10 :: s -> a:b g_10 :: s -> s -> a:b hole_c1_31 :: c hole_s2_31 :: s hole_c13_31 :: c1 hole_c2:c34_31 :: c2:c3 hole_c45_31 :: c4 hole_c5:c66_31 :: c5:c6 hole_c77_31 :: c7 hole_c8:c98_31 :: c8:c9 hole_c109_31 :: c10 hole_c11:c1210_31 :: c11:c12 hole_c1311_31 :: c13 hole_c14:c1512_31 :: c14:c15 hole_c1613_31 :: c16 hole_c17:c1814_31 :: c17:c18 hole_c1915_31 :: c19 hole_c20:c2116_31 :: c20:c21 hole_c2217_31 :: c22 hole_c23:c2418_31 :: c23:c24 hole_c2519_31 :: c25 hole_c26:c2720_31 :: c26:c27 hole_c2821_31 :: c28 hole_c29:c3022_31 :: c29:c30 hole_a:b23_31 :: a:b gen_s24_31 :: Nat -> s gen_c2:c325_31 :: Nat -> c2:c3 gen_c5:c626_31 :: Nat -> c5:c6 gen_c8:c927_31 :: Nat -> c8:c9 gen_c11:c1228_31 :: Nat -> c11:c12 gen_c14:c1529_31 :: Nat -> c14:c15 gen_c17:c1830_31 :: Nat -> c17:c18 gen_c20:c2131_31 :: Nat -> c20:c21 gen_c23:c2432_31 :: Nat -> c23:c24 gen_c26:c2733_31 :: Nat -> c26:c27 gen_c29:c3034_31 :: Nat -> c29:c30 gen_a:b35_31 :: Nat -> a:b Lemmas: G_1(gen_s24_31(+(1, n37_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n37_31) G_2(gen_s24_31(+(1, n1593_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n1593_31) G_3(gen_s24_31(+(1, n4005_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n4005_31) G_4(gen_s24_31(+(1, n6737_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n6737_31) G_5(gen_s24_31(+(1, n9789_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n9789_31) G_6(gen_s24_31(+(1, n13161_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n13161_31) G_7(gen_s24_31(+(1, n16853_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n16853_31) Generator Equations: gen_s24_31(0) <=> hole_s2_31 gen_s24_31(+(x, 1)) <=> s(gen_s24_31(x)) gen_c2:c325_31(0) <=> c2(c) gen_c2:c325_31(+(x, 1)) <=> c3(gen_c2:c325_31(x)) gen_c5:c626_31(0) <=> c5(c1(c2(c))) gen_c5:c626_31(+(x, 1)) <=> c6(gen_c5:c626_31(x)) gen_c8:c927_31(0) <=> c8(c4(c5(c1(c2(c))))) gen_c8:c927_31(+(x, 1)) <=> c9(gen_c8:c927_31(x)) gen_c11:c1228_31(0) <=> c11(c7(c8(c4(c5(c1(c2(c))))))) gen_c11:c1228_31(+(x, 1)) <=> c12(gen_c11:c1228_31(x)) gen_c14:c1529_31(0) <=> c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))) gen_c14:c1529_31(+(x, 1)) <=> c15(gen_c14:c1529_31(x)) gen_c17:c1830_31(0) <=> c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))) gen_c17:c1830_31(+(x, 1)) <=> c18(gen_c17:c1830_31(x)) gen_c20:c2131_31(0) <=> c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))) gen_c20:c2131_31(+(x, 1)) <=> c21(gen_c20:c2131_31(x)) gen_c23:c2432_31(0) <=> c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))) gen_c23:c2432_31(+(x, 1)) <=> c24(gen_c23:c2432_31(x)) gen_c26:c2733_31(0) <=> c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))) gen_c26:c2733_31(+(x, 1)) <=> c27(gen_c26:c2733_31(x)) gen_c29:c3034_31(0) <=> c29(c25(c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))))) gen_c29:c3034_31(+(x, 1)) <=> c30(gen_c29:c3034_31(x)) gen_a:b35_31(0) <=> a gen_a:b35_31(+(x, 1)) <=> b(a, gen_a:b35_31(x)) The following defined symbols remain to be analysed: G_8, G_9, G_10, g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 ---------------------------------------- (65) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: G_8(gen_s24_31(+(1, n20865_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n20865_31) Induction Base: G_8(gen_s24_31(+(1, 0)), gen_s24_31(b)) Induction Step: G_8(gen_s24_31(+(1, +(n20865_31, 1))), gen_s24_31(b)) ->_R^Omega(1) c24(G_8(gen_s24_31(+(1, n20865_31)), gen_s24_31(b))) ->_IH c24(*36_31) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (66) Obligation: Innermost TRS: Rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Types: F_0 :: s -> c c :: c F_1 :: s -> c1 c1 :: c2:c3 -> c1 G_1 :: s -> s -> c2:c3 s :: s -> s c2 :: c -> c2:c3 c3 :: c2:c3 -> c2:c3 F_2 :: s -> c4 c4 :: c5:c6 -> c4 G_2 :: s -> s -> c5:c6 c5 :: c1 -> c5:c6 c6 :: c5:c6 -> c5:c6 F_3 :: s -> c7 c7 :: c8:c9 -> c7 G_3 :: s -> s -> c8:c9 c8 :: c4 -> c8:c9 c9 :: c8:c9 -> c8:c9 F_4 :: s -> c10 c10 :: c11:c12 -> c10 G_4 :: s -> s -> c11:c12 c11 :: c7 -> c11:c12 c12 :: c11:c12 -> c11:c12 F_5 :: s -> c13 c13 :: c14:c15 -> c13 G_5 :: s -> s -> c14:c15 c14 :: c10 -> c14:c15 c15 :: c14:c15 -> c14:c15 F_6 :: s -> c16 c16 :: c17:c18 -> c16 G_6 :: s -> s -> c17:c18 c17 :: c13 -> c17:c18 c18 :: c17:c18 -> c17:c18 F_7 :: s -> c19 c19 :: c20:c21 -> c19 G_7 :: s -> s -> c20:c21 c20 :: c16 -> c20:c21 c21 :: c20:c21 -> c20:c21 F_8 :: s -> c22 c22 :: c23:c24 -> c22 G_8 :: s -> s -> c23:c24 c23 :: c19 -> c23:c24 c24 :: c23:c24 -> c23:c24 F_9 :: s -> c25 c25 :: c26:c27 -> c25 G_9 :: s -> s -> c26:c27 c26 :: c22 -> c26:c27 c27 :: c26:c27 -> c26:c27 F_10 :: s -> c28 c28 :: c29:c30 -> c28 G_10 :: s -> s -> c29:c30 c29 :: c25 -> c29:c30 c30 :: c29:c30 -> c29:c30 f_0 :: s -> a:b a :: a:b f_1 :: s -> a:b g_1 :: s -> s -> a:b b :: a:b -> a:b -> a:b f_2 :: s -> a:b g_2 :: s -> s -> a:b f_3 :: s -> a:b g_3 :: s -> s -> a:b f_4 :: s -> a:b g_4 :: s -> s -> a:b f_5 :: s -> a:b g_5 :: s -> s -> a:b f_6 :: s -> a:b g_6 :: s -> s -> a:b f_7 :: s -> a:b g_7 :: s -> s -> a:b f_8 :: s -> a:b g_8 :: s -> s -> a:b f_9 :: s -> a:b g_9 :: s -> s -> a:b f_10 :: s -> a:b g_10 :: s -> s -> a:b hole_c1_31 :: c hole_s2_31 :: s hole_c13_31 :: c1 hole_c2:c34_31 :: c2:c3 hole_c45_31 :: c4 hole_c5:c66_31 :: c5:c6 hole_c77_31 :: c7 hole_c8:c98_31 :: c8:c9 hole_c109_31 :: c10 hole_c11:c1210_31 :: c11:c12 hole_c1311_31 :: c13 hole_c14:c1512_31 :: c14:c15 hole_c1613_31 :: c16 hole_c17:c1814_31 :: c17:c18 hole_c1915_31 :: c19 hole_c20:c2116_31 :: c20:c21 hole_c2217_31 :: c22 hole_c23:c2418_31 :: c23:c24 hole_c2519_31 :: c25 hole_c26:c2720_31 :: c26:c27 hole_c2821_31 :: c28 hole_c29:c3022_31 :: c29:c30 hole_a:b23_31 :: a:b gen_s24_31 :: Nat -> s gen_c2:c325_31 :: Nat -> c2:c3 gen_c5:c626_31 :: Nat -> c5:c6 gen_c8:c927_31 :: Nat -> c8:c9 gen_c11:c1228_31 :: Nat -> c11:c12 gen_c14:c1529_31 :: Nat -> c14:c15 gen_c17:c1830_31 :: Nat -> c17:c18 gen_c20:c2131_31 :: Nat -> c20:c21 gen_c23:c2432_31 :: Nat -> c23:c24 gen_c26:c2733_31 :: Nat -> c26:c27 gen_c29:c3034_31 :: Nat -> c29:c30 gen_a:b35_31 :: Nat -> a:b Lemmas: G_1(gen_s24_31(+(1, n37_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n37_31) G_2(gen_s24_31(+(1, n1593_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n1593_31) G_3(gen_s24_31(+(1, n4005_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n4005_31) G_4(gen_s24_31(+(1, n6737_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n6737_31) G_5(gen_s24_31(+(1, n9789_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n9789_31) G_6(gen_s24_31(+(1, n13161_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n13161_31) G_7(gen_s24_31(+(1, n16853_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n16853_31) G_8(gen_s24_31(+(1, n20865_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n20865_31) Generator Equations: gen_s24_31(0) <=> hole_s2_31 gen_s24_31(+(x, 1)) <=> s(gen_s24_31(x)) gen_c2:c325_31(0) <=> c2(c) gen_c2:c325_31(+(x, 1)) <=> c3(gen_c2:c325_31(x)) gen_c5:c626_31(0) <=> c5(c1(c2(c))) gen_c5:c626_31(+(x, 1)) <=> c6(gen_c5:c626_31(x)) gen_c8:c927_31(0) <=> c8(c4(c5(c1(c2(c))))) gen_c8:c927_31(+(x, 1)) <=> c9(gen_c8:c927_31(x)) gen_c11:c1228_31(0) <=> c11(c7(c8(c4(c5(c1(c2(c))))))) gen_c11:c1228_31(+(x, 1)) <=> c12(gen_c11:c1228_31(x)) gen_c14:c1529_31(0) <=> c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))) gen_c14:c1529_31(+(x, 1)) <=> c15(gen_c14:c1529_31(x)) gen_c17:c1830_31(0) <=> c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))) gen_c17:c1830_31(+(x, 1)) <=> c18(gen_c17:c1830_31(x)) gen_c20:c2131_31(0) <=> c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))) gen_c20:c2131_31(+(x, 1)) <=> c21(gen_c20:c2131_31(x)) gen_c23:c2432_31(0) <=> c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))) gen_c23:c2432_31(+(x, 1)) <=> c24(gen_c23:c2432_31(x)) gen_c26:c2733_31(0) <=> c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))) gen_c26:c2733_31(+(x, 1)) <=> c27(gen_c26:c2733_31(x)) gen_c29:c3034_31(0) <=> c29(c25(c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))))) gen_c29:c3034_31(+(x, 1)) <=> c30(gen_c29:c3034_31(x)) gen_a:b35_31(0) <=> a gen_a:b35_31(+(x, 1)) <=> b(a, gen_a:b35_31(x)) The following defined symbols remain to be analysed: G_9, G_10, g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 ---------------------------------------- (67) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: G_9(gen_s24_31(+(1, n25197_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n25197_31) Induction Base: G_9(gen_s24_31(+(1, 0)), gen_s24_31(b)) Induction Step: G_9(gen_s24_31(+(1, +(n25197_31, 1))), gen_s24_31(b)) ->_R^Omega(1) c27(G_9(gen_s24_31(+(1, n25197_31)), gen_s24_31(b))) ->_IH c27(*36_31) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (68) Obligation: Innermost TRS: Rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Types: F_0 :: s -> c c :: c F_1 :: s -> c1 c1 :: c2:c3 -> c1 G_1 :: s -> s -> c2:c3 s :: s -> s c2 :: c -> c2:c3 c3 :: c2:c3 -> c2:c3 F_2 :: s -> c4 c4 :: c5:c6 -> c4 G_2 :: s -> s -> c5:c6 c5 :: c1 -> c5:c6 c6 :: c5:c6 -> c5:c6 F_3 :: s -> c7 c7 :: c8:c9 -> c7 G_3 :: s -> s -> c8:c9 c8 :: c4 -> c8:c9 c9 :: c8:c9 -> c8:c9 F_4 :: s -> c10 c10 :: c11:c12 -> c10 G_4 :: s -> s -> c11:c12 c11 :: c7 -> c11:c12 c12 :: c11:c12 -> c11:c12 F_5 :: s -> c13 c13 :: c14:c15 -> c13 G_5 :: s -> s -> c14:c15 c14 :: c10 -> c14:c15 c15 :: c14:c15 -> c14:c15 F_6 :: s -> c16 c16 :: c17:c18 -> c16 G_6 :: s -> s -> c17:c18 c17 :: c13 -> c17:c18 c18 :: c17:c18 -> c17:c18 F_7 :: s -> c19 c19 :: c20:c21 -> c19 G_7 :: s -> s -> c20:c21 c20 :: c16 -> c20:c21 c21 :: c20:c21 -> c20:c21 F_8 :: s -> c22 c22 :: c23:c24 -> c22 G_8 :: s -> s -> c23:c24 c23 :: c19 -> c23:c24 c24 :: c23:c24 -> c23:c24 F_9 :: s -> c25 c25 :: c26:c27 -> c25 G_9 :: s -> s -> c26:c27 c26 :: c22 -> c26:c27 c27 :: c26:c27 -> c26:c27 F_10 :: s -> c28 c28 :: c29:c30 -> c28 G_10 :: s -> s -> c29:c30 c29 :: c25 -> c29:c30 c30 :: c29:c30 -> c29:c30 f_0 :: s -> a:b a :: a:b f_1 :: s -> a:b g_1 :: s -> s -> a:b b :: a:b -> a:b -> a:b f_2 :: s -> a:b g_2 :: s -> s -> a:b f_3 :: s -> a:b g_3 :: s -> s -> a:b f_4 :: s -> a:b g_4 :: s -> s -> a:b f_5 :: s -> a:b g_5 :: s -> s -> a:b f_6 :: s -> a:b g_6 :: s -> s -> a:b f_7 :: s -> a:b g_7 :: s -> s -> a:b f_8 :: s -> a:b g_8 :: s -> s -> a:b f_9 :: s -> a:b g_9 :: s -> s -> a:b f_10 :: s -> a:b g_10 :: s -> s -> a:b hole_c1_31 :: c hole_s2_31 :: s hole_c13_31 :: c1 hole_c2:c34_31 :: c2:c3 hole_c45_31 :: c4 hole_c5:c66_31 :: c5:c6 hole_c77_31 :: c7 hole_c8:c98_31 :: c8:c9 hole_c109_31 :: c10 hole_c11:c1210_31 :: c11:c12 hole_c1311_31 :: c13 hole_c14:c1512_31 :: c14:c15 hole_c1613_31 :: c16 hole_c17:c1814_31 :: c17:c18 hole_c1915_31 :: c19 hole_c20:c2116_31 :: c20:c21 hole_c2217_31 :: c22 hole_c23:c2418_31 :: c23:c24 hole_c2519_31 :: c25 hole_c26:c2720_31 :: c26:c27 hole_c2821_31 :: c28 hole_c29:c3022_31 :: c29:c30 hole_a:b23_31 :: a:b gen_s24_31 :: Nat -> s gen_c2:c325_31 :: Nat -> c2:c3 gen_c5:c626_31 :: Nat -> c5:c6 gen_c8:c927_31 :: Nat -> c8:c9 gen_c11:c1228_31 :: Nat -> c11:c12 gen_c14:c1529_31 :: Nat -> c14:c15 gen_c17:c1830_31 :: Nat -> c17:c18 gen_c20:c2131_31 :: Nat -> c20:c21 gen_c23:c2432_31 :: Nat -> c23:c24 gen_c26:c2733_31 :: Nat -> c26:c27 gen_c29:c3034_31 :: Nat -> c29:c30 gen_a:b35_31 :: Nat -> a:b Lemmas: G_1(gen_s24_31(+(1, n37_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n37_31) G_2(gen_s24_31(+(1, n1593_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n1593_31) G_3(gen_s24_31(+(1, n4005_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n4005_31) G_4(gen_s24_31(+(1, n6737_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n6737_31) G_5(gen_s24_31(+(1, n9789_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n9789_31) G_6(gen_s24_31(+(1, n13161_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n13161_31) G_7(gen_s24_31(+(1, n16853_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n16853_31) G_8(gen_s24_31(+(1, n20865_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n20865_31) G_9(gen_s24_31(+(1, n25197_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n25197_31) Generator Equations: gen_s24_31(0) <=> hole_s2_31 gen_s24_31(+(x, 1)) <=> s(gen_s24_31(x)) gen_c2:c325_31(0) <=> c2(c) gen_c2:c325_31(+(x, 1)) <=> c3(gen_c2:c325_31(x)) gen_c5:c626_31(0) <=> c5(c1(c2(c))) gen_c5:c626_31(+(x, 1)) <=> c6(gen_c5:c626_31(x)) gen_c8:c927_31(0) <=> c8(c4(c5(c1(c2(c))))) gen_c8:c927_31(+(x, 1)) <=> c9(gen_c8:c927_31(x)) gen_c11:c1228_31(0) <=> c11(c7(c8(c4(c5(c1(c2(c))))))) gen_c11:c1228_31(+(x, 1)) <=> c12(gen_c11:c1228_31(x)) gen_c14:c1529_31(0) <=> c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))) gen_c14:c1529_31(+(x, 1)) <=> c15(gen_c14:c1529_31(x)) gen_c17:c1830_31(0) <=> c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))) gen_c17:c1830_31(+(x, 1)) <=> c18(gen_c17:c1830_31(x)) gen_c20:c2131_31(0) <=> c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))) gen_c20:c2131_31(+(x, 1)) <=> c21(gen_c20:c2131_31(x)) gen_c23:c2432_31(0) <=> c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))) gen_c23:c2432_31(+(x, 1)) <=> c24(gen_c23:c2432_31(x)) gen_c26:c2733_31(0) <=> c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))) gen_c26:c2733_31(+(x, 1)) <=> c27(gen_c26:c2733_31(x)) gen_c29:c3034_31(0) <=> c29(c25(c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))))) gen_c29:c3034_31(+(x, 1)) <=> c30(gen_c29:c3034_31(x)) gen_a:b35_31(0) <=> a gen_a:b35_31(+(x, 1)) <=> b(a, gen_a:b35_31(x)) The following defined symbols remain to be analysed: G_10, g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 ---------------------------------------- (69) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: G_10(gen_s24_31(+(1, n29849_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n29849_31) Induction Base: G_10(gen_s24_31(+(1, 0)), gen_s24_31(b)) Induction Step: G_10(gen_s24_31(+(1, +(n29849_31, 1))), gen_s24_31(b)) ->_R^Omega(1) c30(G_10(gen_s24_31(+(1, n29849_31)), gen_s24_31(b))) ->_IH c30(*36_31) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (70) Obligation: Innermost TRS: Rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Types: F_0 :: s -> c c :: c F_1 :: s -> c1 c1 :: c2:c3 -> c1 G_1 :: s -> s -> c2:c3 s :: s -> s c2 :: c -> c2:c3 c3 :: c2:c3 -> c2:c3 F_2 :: s -> c4 c4 :: c5:c6 -> c4 G_2 :: s -> s -> c5:c6 c5 :: c1 -> c5:c6 c6 :: c5:c6 -> c5:c6 F_3 :: s -> c7 c7 :: c8:c9 -> c7 G_3 :: s -> s -> c8:c9 c8 :: c4 -> c8:c9 c9 :: c8:c9 -> c8:c9 F_4 :: s -> c10 c10 :: c11:c12 -> c10 G_4 :: s -> s -> c11:c12 c11 :: c7 -> c11:c12 c12 :: c11:c12 -> c11:c12 F_5 :: s -> c13 c13 :: c14:c15 -> c13 G_5 :: s -> s -> c14:c15 c14 :: c10 -> c14:c15 c15 :: c14:c15 -> c14:c15 F_6 :: s -> c16 c16 :: c17:c18 -> c16 G_6 :: s -> s -> c17:c18 c17 :: c13 -> c17:c18 c18 :: c17:c18 -> c17:c18 F_7 :: s -> c19 c19 :: c20:c21 -> c19 G_7 :: s -> s -> c20:c21 c20 :: c16 -> c20:c21 c21 :: c20:c21 -> c20:c21 F_8 :: s -> c22 c22 :: c23:c24 -> c22 G_8 :: s -> s -> c23:c24 c23 :: c19 -> c23:c24 c24 :: c23:c24 -> c23:c24 F_9 :: s -> c25 c25 :: c26:c27 -> c25 G_9 :: s -> s -> c26:c27 c26 :: c22 -> c26:c27 c27 :: c26:c27 -> c26:c27 F_10 :: s -> c28 c28 :: c29:c30 -> c28 G_10 :: s -> s -> c29:c30 c29 :: c25 -> c29:c30 c30 :: c29:c30 -> c29:c30 f_0 :: s -> a:b a :: a:b f_1 :: s -> a:b g_1 :: s -> s -> a:b b :: a:b -> a:b -> a:b f_2 :: s -> a:b g_2 :: s -> s -> a:b f_3 :: s -> a:b g_3 :: s -> s -> a:b f_4 :: s -> a:b g_4 :: s -> s -> a:b f_5 :: s -> a:b g_5 :: s -> s -> a:b f_6 :: s -> a:b g_6 :: s -> s -> a:b f_7 :: s -> a:b g_7 :: s -> s -> a:b f_8 :: s -> a:b g_8 :: s -> s -> a:b f_9 :: s -> a:b g_9 :: s -> s -> a:b f_10 :: s -> a:b g_10 :: s -> s -> a:b hole_c1_31 :: c hole_s2_31 :: s hole_c13_31 :: c1 hole_c2:c34_31 :: c2:c3 hole_c45_31 :: c4 hole_c5:c66_31 :: c5:c6 hole_c77_31 :: c7 hole_c8:c98_31 :: c8:c9 hole_c109_31 :: c10 hole_c11:c1210_31 :: c11:c12 hole_c1311_31 :: c13 hole_c14:c1512_31 :: c14:c15 hole_c1613_31 :: c16 hole_c17:c1814_31 :: c17:c18 hole_c1915_31 :: c19 hole_c20:c2116_31 :: c20:c21 hole_c2217_31 :: c22 hole_c23:c2418_31 :: c23:c24 hole_c2519_31 :: c25 hole_c26:c2720_31 :: c26:c27 hole_c2821_31 :: c28 hole_c29:c3022_31 :: c29:c30 hole_a:b23_31 :: a:b gen_s24_31 :: Nat -> s gen_c2:c325_31 :: Nat -> c2:c3 gen_c5:c626_31 :: Nat -> c5:c6 gen_c8:c927_31 :: Nat -> c8:c9 gen_c11:c1228_31 :: Nat -> c11:c12 gen_c14:c1529_31 :: Nat -> c14:c15 gen_c17:c1830_31 :: Nat -> c17:c18 gen_c20:c2131_31 :: Nat -> c20:c21 gen_c23:c2432_31 :: Nat -> c23:c24 gen_c26:c2733_31 :: Nat -> c26:c27 gen_c29:c3034_31 :: Nat -> c29:c30 gen_a:b35_31 :: Nat -> a:b Lemmas: G_1(gen_s24_31(+(1, n37_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n37_31) G_2(gen_s24_31(+(1, n1593_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n1593_31) G_3(gen_s24_31(+(1, n4005_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n4005_31) G_4(gen_s24_31(+(1, n6737_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n6737_31) G_5(gen_s24_31(+(1, n9789_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n9789_31) G_6(gen_s24_31(+(1, n13161_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n13161_31) G_7(gen_s24_31(+(1, n16853_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n16853_31) G_8(gen_s24_31(+(1, n20865_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n20865_31) G_9(gen_s24_31(+(1, n25197_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n25197_31) G_10(gen_s24_31(+(1, n29849_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n29849_31) Generator Equations: gen_s24_31(0) <=> hole_s2_31 gen_s24_31(+(x, 1)) <=> s(gen_s24_31(x)) gen_c2:c325_31(0) <=> c2(c) gen_c2:c325_31(+(x, 1)) <=> c3(gen_c2:c325_31(x)) gen_c5:c626_31(0) <=> c5(c1(c2(c))) gen_c5:c626_31(+(x, 1)) <=> c6(gen_c5:c626_31(x)) gen_c8:c927_31(0) <=> c8(c4(c5(c1(c2(c))))) gen_c8:c927_31(+(x, 1)) <=> c9(gen_c8:c927_31(x)) gen_c11:c1228_31(0) <=> c11(c7(c8(c4(c5(c1(c2(c))))))) gen_c11:c1228_31(+(x, 1)) <=> c12(gen_c11:c1228_31(x)) gen_c14:c1529_31(0) <=> c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))) gen_c14:c1529_31(+(x, 1)) <=> c15(gen_c14:c1529_31(x)) gen_c17:c1830_31(0) <=> c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))) gen_c17:c1830_31(+(x, 1)) <=> c18(gen_c17:c1830_31(x)) gen_c20:c2131_31(0) <=> c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))) gen_c20:c2131_31(+(x, 1)) <=> c21(gen_c20:c2131_31(x)) gen_c23:c2432_31(0) <=> c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))) gen_c23:c2432_31(+(x, 1)) <=> c24(gen_c23:c2432_31(x)) gen_c26:c2733_31(0) <=> c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))) gen_c26:c2733_31(+(x, 1)) <=> c27(gen_c26:c2733_31(x)) gen_c29:c3034_31(0) <=> c29(c25(c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))))) gen_c29:c3034_31(+(x, 1)) <=> c30(gen_c29:c3034_31(x)) gen_a:b35_31(0) <=> a gen_a:b35_31(+(x, 1)) <=> b(a, gen_a:b35_31(x)) The following defined symbols remain to be analysed: g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 ---------------------------------------- (71) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: g_1(gen_s24_31(+(1, n34821_31)), gen_s24_31(b)) -> *36_31, rt in Omega(0) Induction Base: g_1(gen_s24_31(+(1, 0)), gen_s24_31(b)) Induction Step: g_1(gen_s24_31(+(1, +(n34821_31, 1))), gen_s24_31(b)) ->_R^Omega(0) b(f_0(gen_s24_31(b)), g_1(gen_s24_31(+(1, n34821_31)), gen_s24_31(b))) ->_R^Omega(0) b(a, g_1(gen_s24_31(+(1, n34821_31)), gen_s24_31(b))) ->_IH b(a, *36_31) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (72) Obligation: Innermost TRS: Rules: F_0(z0) -> c F_1(z0) -> c1(G_1(z0, z0)) G_1(s(z0), z1) -> c2(F_0(z1)) G_1(s(z0), z1) -> c3(G_1(z0, z1)) F_2(z0) -> c4(G_2(z0, z0)) G_2(s(z0), z1) -> c5(F_1(z1)) G_2(s(z0), z1) -> c6(G_2(z0, z1)) F_3(z0) -> c7(G_3(z0, z0)) G_3(s(z0), z1) -> c8(F_2(z1)) G_3(s(z0), z1) -> c9(G_3(z0, z1)) F_4(z0) -> c10(G_4(z0, z0)) G_4(s(z0), z1) -> c11(F_3(z1)) G_4(s(z0), z1) -> c12(G_4(z0, z1)) F_5(z0) -> c13(G_5(z0, z0)) G_5(s(z0), z1) -> c14(F_4(z1)) G_5(s(z0), z1) -> c15(G_5(z0, z1)) F_6(z0) -> c16(G_6(z0, z0)) G_6(s(z0), z1) -> c17(F_5(z1)) G_6(s(z0), z1) -> c18(G_6(z0, z1)) F_7(z0) -> c19(G_7(z0, z0)) G_7(s(z0), z1) -> c20(F_6(z1)) G_7(s(z0), z1) -> c21(G_7(z0, z1)) F_8(z0) -> c22(G_8(z0, z0)) G_8(s(z0), z1) -> c23(F_7(z1)) G_8(s(z0), z1) -> c24(G_8(z0, z1)) F_9(z0) -> c25(G_9(z0, z0)) G_9(s(z0), z1) -> c26(F_8(z1)) G_9(s(z0), z1) -> c27(G_9(z0, z1)) F_10(z0) -> c28(G_10(z0, z0)) G_10(s(z0), z1) -> c29(F_9(z1)) G_10(s(z0), z1) -> c30(G_10(z0, z1)) f_0(z0) -> a f_1(z0) -> g_1(z0, z0) g_1(s(z0), z1) -> b(f_0(z1), g_1(z0, z1)) f_2(z0) -> g_2(z0, z0) g_2(s(z0), z1) -> b(f_1(z1), g_2(z0, z1)) f_3(z0) -> g_3(z0, z0) g_3(s(z0), z1) -> b(f_2(z1), g_3(z0, z1)) f_4(z0) -> g_4(z0, z0) g_4(s(z0), z1) -> b(f_3(z1), g_4(z0, z1)) f_5(z0) -> g_5(z0, z0) g_5(s(z0), z1) -> b(f_4(z1), g_5(z0, z1)) f_6(z0) -> g_6(z0, z0) g_6(s(z0), z1) -> b(f_5(z1), g_6(z0, z1)) f_7(z0) -> g_7(z0, z0) g_7(s(z0), z1) -> b(f_6(z1), g_7(z0, z1)) f_8(z0) -> g_8(z0, z0) g_8(s(z0), z1) -> b(f_7(z1), g_8(z0, z1)) f_9(z0) -> g_9(z0, z0) g_9(s(z0), z1) -> b(f_8(z1), g_9(z0, z1)) f_10(z0) -> g_10(z0, z0) g_10(s(z0), z1) -> b(f_9(z1), g_10(z0, z1)) Types: F_0 :: s -> c c :: c F_1 :: s -> c1 c1 :: c2:c3 -> c1 G_1 :: s -> s -> c2:c3 s :: s -> s c2 :: c -> c2:c3 c3 :: c2:c3 -> c2:c3 F_2 :: s -> c4 c4 :: c5:c6 -> c4 G_2 :: s -> s -> c5:c6 c5 :: c1 -> c5:c6 c6 :: c5:c6 -> c5:c6 F_3 :: s -> c7 c7 :: c8:c9 -> c7 G_3 :: s -> s -> c8:c9 c8 :: c4 -> c8:c9 c9 :: c8:c9 -> c8:c9 F_4 :: s -> c10 c10 :: c11:c12 -> c10 G_4 :: s -> s -> c11:c12 c11 :: c7 -> c11:c12 c12 :: c11:c12 -> c11:c12 F_5 :: s -> c13 c13 :: c14:c15 -> c13 G_5 :: s -> s -> c14:c15 c14 :: c10 -> c14:c15 c15 :: c14:c15 -> c14:c15 F_6 :: s -> c16 c16 :: c17:c18 -> c16 G_6 :: s -> s -> c17:c18 c17 :: c13 -> c17:c18 c18 :: c17:c18 -> c17:c18 F_7 :: s -> c19 c19 :: c20:c21 -> c19 G_7 :: s -> s -> c20:c21 c20 :: c16 -> c20:c21 c21 :: c20:c21 -> c20:c21 F_8 :: s -> c22 c22 :: c23:c24 -> c22 G_8 :: s -> s -> c23:c24 c23 :: c19 -> c23:c24 c24 :: c23:c24 -> c23:c24 F_9 :: s -> c25 c25 :: c26:c27 -> c25 G_9 :: s -> s -> c26:c27 c26 :: c22 -> c26:c27 c27 :: c26:c27 -> c26:c27 F_10 :: s -> c28 c28 :: c29:c30 -> c28 G_10 :: s -> s -> c29:c30 c29 :: c25 -> c29:c30 c30 :: c29:c30 -> c29:c30 f_0 :: s -> a:b a :: a:b f_1 :: s -> a:b g_1 :: s -> s -> a:b b :: a:b -> a:b -> a:b f_2 :: s -> a:b g_2 :: s -> s -> a:b f_3 :: s -> a:b g_3 :: s -> s -> a:b f_4 :: s -> a:b g_4 :: s -> s -> a:b f_5 :: s -> a:b g_5 :: s -> s -> a:b f_6 :: s -> a:b g_6 :: s -> s -> a:b f_7 :: s -> a:b g_7 :: s -> s -> a:b f_8 :: s -> a:b g_8 :: s -> s -> a:b f_9 :: s -> a:b g_9 :: s -> s -> a:b f_10 :: s -> a:b g_10 :: s -> s -> a:b hole_c1_31 :: c hole_s2_31 :: s hole_c13_31 :: c1 hole_c2:c34_31 :: c2:c3 hole_c45_31 :: c4 hole_c5:c66_31 :: c5:c6 hole_c77_31 :: c7 hole_c8:c98_31 :: c8:c9 hole_c109_31 :: c10 hole_c11:c1210_31 :: c11:c12 hole_c1311_31 :: c13 hole_c14:c1512_31 :: c14:c15 hole_c1613_31 :: c16 hole_c17:c1814_31 :: c17:c18 hole_c1915_31 :: c19 hole_c20:c2116_31 :: c20:c21 hole_c2217_31 :: c22 hole_c23:c2418_31 :: c23:c24 hole_c2519_31 :: c25 hole_c26:c2720_31 :: c26:c27 hole_c2821_31 :: c28 hole_c29:c3022_31 :: c29:c30 hole_a:b23_31 :: a:b gen_s24_31 :: Nat -> s gen_c2:c325_31 :: Nat -> c2:c3 gen_c5:c626_31 :: Nat -> c5:c6 gen_c8:c927_31 :: Nat -> c8:c9 gen_c11:c1228_31 :: Nat -> c11:c12 gen_c14:c1529_31 :: Nat -> c14:c15 gen_c17:c1830_31 :: Nat -> c17:c18 gen_c20:c2131_31 :: Nat -> c20:c21 gen_c23:c2432_31 :: Nat -> c23:c24 gen_c26:c2733_31 :: Nat -> c26:c27 gen_c29:c3034_31 :: Nat -> c29:c30 gen_a:b35_31 :: Nat -> a:b Lemmas: G_1(gen_s24_31(+(1, n37_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n37_31) G_2(gen_s24_31(+(1, n1593_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n1593_31) G_3(gen_s24_31(+(1, n4005_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n4005_31) G_4(gen_s24_31(+(1, n6737_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n6737_31) G_5(gen_s24_31(+(1, n9789_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n9789_31) G_6(gen_s24_31(+(1, n13161_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n13161_31) G_7(gen_s24_31(+(1, n16853_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n16853_31) G_8(gen_s24_31(+(1, n20865_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n20865_31) G_9(gen_s24_31(+(1, n25197_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n25197_31) G_10(gen_s24_31(+(1, n29849_31)), gen_s24_31(b)) -> *36_31, rt in Omega(n29849_31) g_1(gen_s24_31(+(1, n34821_31)), gen_s24_31(b)) -> *36_31, rt in Omega(0) Generator Equations: gen_s24_31(0) <=> hole_s2_31 gen_s24_31(+(x, 1)) <=> s(gen_s24_31(x)) gen_c2:c325_31(0) <=> c2(c) gen_c2:c325_31(+(x, 1)) <=> c3(gen_c2:c325_31(x)) gen_c5:c626_31(0) <=> c5(c1(c2(c))) gen_c5:c626_31(+(x, 1)) <=> c6(gen_c5:c626_31(x)) gen_c8:c927_31(0) <=> c8(c4(c5(c1(c2(c))))) gen_c8:c927_31(+(x, 1)) <=> c9(gen_c8:c927_31(x)) gen_c11:c1228_31(0) <=> c11(c7(c8(c4(c5(c1(c2(c))))))) gen_c11:c1228_31(+(x, 1)) <=> c12(gen_c11:c1228_31(x)) gen_c14:c1529_31(0) <=> c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))) gen_c14:c1529_31(+(x, 1)) <=> c15(gen_c14:c1529_31(x)) gen_c17:c1830_31(0) <=> c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))) gen_c17:c1830_31(+(x, 1)) <=> c18(gen_c17:c1830_31(x)) gen_c20:c2131_31(0) <=> c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))) gen_c20:c2131_31(+(x, 1)) <=> c21(gen_c20:c2131_31(x)) gen_c23:c2432_31(0) <=> c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))) gen_c23:c2432_31(+(x, 1)) <=> c24(gen_c23:c2432_31(x)) gen_c26:c2733_31(0) <=> c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))) gen_c26:c2733_31(+(x, 1)) <=> c27(gen_c26:c2733_31(x)) gen_c29:c3034_31(0) <=> c29(c25(c26(c22(c23(c19(c20(c16(c17(c13(c14(c10(c11(c7(c8(c4(c5(c1(c2(c))))))))))))))))))) gen_c29:c3034_31(+(x, 1)) <=> c30(gen_c29:c3034_31(x)) gen_a:b35_31(0) <=> a gen_a:b35_31(+(x, 1)) <=> b(a, gen_a:b35_31(x)) The following defined symbols remain to be analysed: g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10