WORST_CASE(Omega(n^1),O(n^2)) proof of /export/starexec/sandbox/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 966 ms] (2) CpxRelTRS (3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (4) CdtProblem (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CdtProblem (7) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CdtProblem (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CdtProblem (13) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CdtProblem (15) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CdtProblem (17) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CdtProblem (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 27 ms] (20) CdtProblem (21) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (22) CdtProblem (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 61 ms] (40) CdtProblem (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 29 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 40 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 28 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 30 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (104) CdtProblem (105) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 43 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (120) CdtProblem (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (124) CdtProblem (125) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 11 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 32 ms] (134) CdtProblem (135) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 3260 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 10 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 3 ms] (148) CdtProblem (149) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 36 ms] (150) CdtProblem (151) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 27 ms] (152) CdtProblem (153) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 1098 ms] (154) CdtProblem (155) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (156) BOUNDS(1, 1) (157) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CpxRelTRS (159) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (160) typed CpxTrs (161) OrderProof [LOWER BOUND(ID), 0 ms] (162) typed CpxTrs (163) RewriteLemmaProof [LOWER BOUND(ID), 213 ms] (164) typed CpxTrs (165) RewriteLemmaProof [LOWER BOUND(ID), 119 ms] (166) BEST (167) proven lower bound (168) LowerBoundPropagationProof [FINISHED, 0 ms] (169) BOUNDS(n^1, INF) (170) typed CpxTrs (171) RewriteLemmaProof [LOWER BOUND(ID), 51 ms] (172) typed CpxTrs (173) RewriteLemmaProof [LOWER BOUND(ID), 92 ms] (174) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0)) COND2(true, z0) -> c1(COND1(neq(z0, 0), div2(z0)), NEQ(z0, 0)) COND2(true, z0) -> c2(COND1(neq(z0, 0), div2(z0)), DIV2(z0)) COND2(false, z0) -> c3(COND1(neq(z0, 0), p(z0)), NEQ(z0, 0)) COND2(false, z0) -> c4(COND1(neq(z0, 0), p(z0)), P(z0)) NEQ(0, 0) -> c5 NEQ(0, s(z0)) -> c6 NEQ(s(z0), 0) -> c7 NEQ(s(z0), s(y)) -> c8(NEQ(z0, y)) EVEN(0) -> c9 EVEN(s(0)) -> c10 EVEN(s(s(z0))) -> c11(EVEN(z0)) DIV2(0) -> c12 DIV2(s(0)) -> c13 DIV2(s(s(z0))) -> c14(DIV2(z0)) P(0) -> c15 P(s(z0)) -> c16 The (relative) TRS S consists of the following rules: cond1(true, z0) -> cond2(even(z0), z0) cond2(true, z0) -> cond1(neq(z0, 0), div2(z0)) cond2(false, z0) -> cond1(neq(z0, 0), p(z0)) neq(0, 0) -> false neq(0, s(z0)) -> true neq(s(z0), 0) -> true neq(s(z0), s(y)) -> neq(z0, y) even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0)) COND2(true, z0) -> c1(COND1(neq(z0, 0), div2(z0)), NEQ(z0, 0)) COND2(true, z0) -> c2(COND1(neq(z0, 0), div2(z0)), DIV2(z0)) COND2(false, z0) -> c3(COND1(neq(z0, 0), p(z0)), NEQ(z0, 0)) COND2(false, z0) -> c4(COND1(neq(z0, 0), p(z0)), P(z0)) NEQ(0, 0) -> c5 NEQ(0, s(z0)) -> c6 NEQ(s(z0), 0) -> c7 NEQ(s(z0), s(y)) -> c8(NEQ(z0, y)) EVEN(0) -> c9 EVEN(s(0)) -> c10 EVEN(s(s(z0))) -> c11(EVEN(z0)) DIV2(0) -> c12 DIV2(s(0)) -> c13 DIV2(s(s(z0))) -> c14(DIV2(z0)) P(0) -> c15 P(s(z0)) -> c16 The (relative) TRS S consists of the following rules: cond1(true, z0) -> cond2(even(z0), z0) cond2(true, z0) -> cond1(neq(z0, 0), div2(z0)) cond2(false, z0) -> cond1(neq(z0, 0), p(z0)) neq(0, 0) -> false neq(0, s(z0)) -> true neq(s(z0), 0) -> true neq(s(z0), s(y)) -> neq(z0, y) even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Rewrite Strategy: INNERMOST ---------------------------------------- (3) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (4) Obligation: Complexity Dependency Tuples Problem Rules: cond1(true, z0) -> cond2(even(z0), z0) cond2(true, z0) -> cond1(neq(z0, 0), div2(z0)) cond2(false, z0) -> cond1(neq(z0, 0), p(z0)) neq(0, 0) -> false neq(0, s(z0)) -> true neq(s(z0), 0) -> true neq(s(z0), s(y)) -> neq(z0, y) even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0)) COND2(true, z0) -> c1(COND1(neq(z0, 0), div2(z0)), NEQ(z0, 0)) COND2(true, z0) -> c2(COND1(neq(z0, 0), div2(z0)), DIV2(z0)) COND2(false, z0) -> c3(COND1(neq(z0, 0), p(z0)), NEQ(z0, 0)) COND2(false, z0) -> c4(COND1(neq(z0, 0), p(z0)), P(z0)) NEQ(0, 0) -> c5 NEQ(0, s(z0)) -> c6 NEQ(s(z0), 0) -> c7 NEQ(s(z0), s(y)) -> c8(NEQ(z0, y)) EVEN(0) -> c9 EVEN(s(0)) -> c10 EVEN(s(s(z0))) -> c11(EVEN(z0)) DIV2(0) -> c12 DIV2(s(0)) -> c13 DIV2(s(s(z0))) -> c14(DIV2(z0)) P(0) -> c15 P(s(z0)) -> c16 Tuples: COND1'(true, z0) -> c17(COND2'(even(z0), z0), EVEN'(z0)) COND2'(true, z0) -> c18(COND1'(neq(z0, 0), div2(z0)), NEQ'(z0, 0), DIV2'(z0)) COND2'(false, z0) -> c19(COND1'(neq(z0, 0), p(z0)), NEQ'(z0, 0), P'(z0)) NEQ'(0, 0) -> c20 NEQ'(0, s(z0)) -> c21 NEQ'(s(z0), 0) -> c22 NEQ'(s(z0), s(y)) -> c23(NEQ'(z0, y)) EVEN'(0) -> c24 EVEN'(s(0)) -> c25 EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(0) -> c27 DIV2'(s(0)) -> c28 DIV2'(s(s(z0))) -> c29(DIV2'(z0)) P'(0) -> c30 P'(s(z0)) -> c31 COND1''(true, z0) -> c32(COND2''(even(z0), z0), EVEN'(z0), EVEN''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), NEQ'(z0, 0), DIV2'(z0), NEQ''(z0, 0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), NEQ'(z0, 0), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0)), NEQ'(z0, 0), P'(z0), NEQ''(z0, 0)) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0)), NEQ'(z0, 0), P'(z0), P''(z0)) NEQ''(0, 0) -> c37 NEQ''(0, s(z0)) -> c38 NEQ''(s(z0), 0) -> c39 NEQ''(s(z0), s(y)) -> c40(NEQ''(z0, y)) EVEN''(0) -> c41 EVEN''(s(0)) -> c42 EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(0) -> c44 DIV2''(s(0)) -> c45 DIV2''(s(s(z0))) -> c46(DIV2''(z0)) P''(0) -> c47 P''(s(z0)) -> c48 S tuples: COND1''(true, z0) -> c32(COND2''(even(z0), z0), EVEN'(z0), EVEN''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), NEQ'(z0, 0), DIV2'(z0), NEQ''(z0, 0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), NEQ'(z0, 0), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0)), NEQ'(z0, 0), P'(z0), NEQ''(z0, 0)) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0)), NEQ'(z0, 0), P'(z0), P''(z0)) NEQ''(0, 0) -> c37 NEQ''(0, s(z0)) -> c38 NEQ''(s(z0), 0) -> c39 NEQ''(s(z0), s(y)) -> c40(NEQ''(z0, y)) EVEN''(0) -> c41 EVEN''(s(0)) -> c42 EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(0) -> c44 DIV2''(s(0)) -> c45 DIV2''(s(s(z0))) -> c46(DIV2''(z0)) P''(0) -> c47 P''(s(z0)) -> c48 K tuples:none Defined Rule Symbols: COND1_2, COND2_2, NEQ_2, EVEN_1, DIV2_1, P_1, cond1_2, cond2_2, neq_2, even_1, div2_1, p_1 Defined Pair Symbols: COND1'_2, COND2'_2, NEQ'_2, EVEN'_1, DIV2'_1, P'_1, COND1''_2, COND2''_2, NEQ''_2, EVEN''_1, DIV2''_1, P''_1 Compound Symbols: c17_2, c18_3, c19_3, c20, c21, c22, c23_1, c24, c25, c26_1, c27, c28, c29_1, c30, c31, c32_3, c33_4, c34_4, c35_4, c36_4, c37, c38, c39, c40_1, c41, c42, c43_1, c44, c45, c46_1, c47, c48 ---------------------------------------- (5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 20 trailing nodes: DIV2''(0) -> c44 NEQ''(0, 0) -> c37 DIV2'(s(0)) -> c28 DIV2''(s(0)) -> c45 EVEN'(s(0)) -> c25 P'(0) -> c30 NEQ''(0, s(z0)) -> c38 P'(s(z0)) -> c31 NEQ'(s(z0), s(y)) -> c23(NEQ'(z0, y)) EVEN''(0) -> c41 DIV2'(0) -> c27 NEQ'(0, s(z0)) -> c21 NEQ'(0, 0) -> c20 NEQ''(s(z0), s(y)) -> c40(NEQ''(z0, y)) EVEN''(s(0)) -> c42 NEQ''(s(z0), 0) -> c39 NEQ'(s(z0), 0) -> c22 P''(s(z0)) -> c48 EVEN'(0) -> c24 P''(0) -> c47 ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: cond1(true, z0) -> cond2(even(z0), z0) cond2(true, z0) -> cond1(neq(z0, 0), div2(z0)) cond2(false, z0) -> cond1(neq(z0, 0), p(z0)) neq(0, 0) -> false neq(0, s(z0)) -> true neq(s(z0), 0) -> true neq(s(z0), s(y)) -> neq(z0, y) even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0)) COND2(true, z0) -> c1(COND1(neq(z0, 0), div2(z0)), NEQ(z0, 0)) COND2(true, z0) -> c2(COND1(neq(z0, 0), div2(z0)), DIV2(z0)) COND2(false, z0) -> c3(COND1(neq(z0, 0), p(z0)), NEQ(z0, 0)) COND2(false, z0) -> c4(COND1(neq(z0, 0), p(z0)), P(z0)) NEQ(0, 0) -> c5 NEQ(0, s(z0)) -> c6 NEQ(s(z0), 0) -> c7 NEQ(s(z0), s(y)) -> c8(NEQ(z0, y)) EVEN(0) -> c9 EVEN(s(0)) -> c10 EVEN(s(s(z0))) -> c11(EVEN(z0)) DIV2(0) -> c12 DIV2(s(0)) -> c13 DIV2(s(s(z0))) -> c14(DIV2(z0)) P(0) -> c15 P(s(z0)) -> c16 Tuples: COND1'(true, z0) -> c17(COND2'(even(z0), z0), EVEN'(z0)) COND2'(true, z0) -> c18(COND1'(neq(z0, 0), div2(z0)), NEQ'(z0, 0), DIV2'(z0)) COND2'(false, z0) -> c19(COND1'(neq(z0, 0), p(z0)), NEQ'(z0, 0), P'(z0)) EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) COND1''(true, z0) -> c32(COND2''(even(z0), z0), EVEN'(z0), EVEN''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), NEQ'(z0, 0), DIV2'(z0), NEQ''(z0, 0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), NEQ'(z0, 0), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0)), NEQ'(z0, 0), P'(z0), NEQ''(z0, 0)) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0)), NEQ'(z0, 0), P'(z0), P''(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) S tuples: COND1''(true, z0) -> c32(COND2''(even(z0), z0), EVEN'(z0), EVEN''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), NEQ'(z0, 0), DIV2'(z0), NEQ''(z0, 0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), NEQ'(z0, 0), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0)), NEQ'(z0, 0), P'(z0), NEQ''(z0, 0)) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0)), NEQ'(z0, 0), P'(z0), P''(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) K tuples:none Defined Rule Symbols: COND1_2, COND2_2, NEQ_2, EVEN_1, DIV2_1, P_1, cond1_2, cond2_2, neq_2, even_1, div2_1, p_1 Defined Pair Symbols: COND1'_2, COND2'_2, EVEN'_1, DIV2'_1, COND1''_2, COND2''_2, EVEN''_1, DIV2''_1 Compound Symbols: c17_2, c18_3, c19_3, c26_1, c29_1, c32_3, c33_4, c34_4, c35_4, c36_4, c43_1, c46_1 ---------------------------------------- (7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing tuple parts ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: cond1(true, z0) -> cond2(even(z0), z0) cond2(true, z0) -> cond1(neq(z0, 0), div2(z0)) cond2(false, z0) -> cond1(neq(z0, 0), p(z0)) neq(0, 0) -> false neq(0, s(z0)) -> true neq(s(z0), 0) -> true neq(s(z0), s(y)) -> neq(z0, y) even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0)) COND2(true, z0) -> c1(COND1(neq(z0, 0), div2(z0)), NEQ(z0, 0)) COND2(true, z0) -> c2(COND1(neq(z0, 0), div2(z0)), DIV2(z0)) COND2(false, z0) -> c3(COND1(neq(z0, 0), p(z0)), NEQ(z0, 0)) COND2(false, z0) -> c4(COND1(neq(z0, 0), p(z0)), P(z0)) NEQ(0, 0) -> c5 NEQ(0, s(z0)) -> c6 NEQ(s(z0), 0) -> c7 NEQ(s(z0), s(y)) -> c8(NEQ(z0, y)) EVEN(0) -> c9 EVEN(s(0)) -> c10 EVEN(s(s(z0))) -> c11(EVEN(z0)) DIV2(0) -> c12 DIV2(s(0)) -> c13 DIV2(s(s(z0))) -> c14(DIV2(z0)) P(0) -> c15 P(s(z0)) -> c16 Tuples: COND1'(true, z0) -> c17(COND2'(even(z0), z0), EVEN'(z0)) EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) COND1''(true, z0) -> c32(COND2''(even(z0), z0), EVEN'(z0), EVEN''(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2'(true, z0) -> c18(COND1'(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2'(false, z0) -> c19(COND1'(neq(z0, 0), p(z0))) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) S tuples: COND1''(true, z0) -> c32(COND2''(even(z0), z0), EVEN'(z0), EVEN''(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) K tuples:none Defined Rule Symbols: COND1_2, COND2_2, NEQ_2, EVEN_1, DIV2_1, P_1, cond1_2, cond2_2, neq_2, even_1, div2_1, p_1 Defined Pair Symbols: COND1'_2, EVEN'_1, DIV2'_1, COND1''_2, EVEN''_1, DIV2''_1, COND2'_2, COND2''_2 Compound Symbols: c17_2, c26_1, c29_1, c32_3, c43_1, c46_1, c18_2, c19_1, c33_2, c34_3, c35_1, c36_1 ---------------------------------------- (9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: cond1(true, z0) -> cond2(even(z0), z0) cond2(true, z0) -> cond1(neq(z0, 0), div2(z0)) cond2(false, z0) -> cond1(neq(z0, 0), p(z0)) neq(0, s(z0)) -> true neq(s(z0), s(y)) -> neq(z0, y) COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0)) COND2(true, z0) -> c1(COND1(neq(z0, 0), div2(z0)), NEQ(z0, 0)) COND2(true, z0) -> c2(COND1(neq(z0, 0), div2(z0)), DIV2(z0)) COND2(false, z0) -> c3(COND1(neq(z0, 0), p(z0)), NEQ(z0, 0)) COND2(false, z0) -> c4(COND1(neq(z0, 0), p(z0)), P(z0)) NEQ(0, 0) -> c5 NEQ(0, s(z0)) -> c6 NEQ(s(z0), 0) -> c7 NEQ(s(z0), s(y)) -> c8(NEQ(z0, y)) EVEN(0) -> c9 EVEN(s(0)) -> c10 EVEN(s(s(z0))) -> c11(EVEN(z0)) DIV2(0) -> c12 DIV2(s(0)) -> c13 DIV2(s(s(z0))) -> c14(DIV2(z0)) P(0) -> c15 P(s(z0)) -> c16 ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: COND1'(true, z0) -> c17(COND2'(even(z0), z0), EVEN'(z0)) EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) COND1''(true, z0) -> c32(COND2''(even(z0), z0), EVEN'(z0), EVEN''(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2'(true, z0) -> c18(COND1'(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2'(false, z0) -> c19(COND1'(neq(z0, 0), p(z0))) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) S tuples: COND1''(true, z0) -> c32(COND2''(even(z0), z0), EVEN'(z0), EVEN''(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) K tuples:none Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: COND1'_2, EVEN'_1, DIV2'_1, COND1''_2, EVEN''_1, DIV2''_1, COND2'_2, COND2''_2 Compound Symbols: c17_2, c26_1, c29_1, c32_3, c43_1, c46_1, c18_2, c19_1, c33_2, c34_3, c35_1, c36_1 ---------------------------------------- (11) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND1'(true, z0) -> c17(COND2'(even(z0), z0), EVEN'(z0)) by COND1'(true, 0) -> c17(COND2'(true, 0), EVEN'(0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0)), EVEN'(s(0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) COND1''(true, z0) -> c32(COND2''(even(z0), z0), EVEN'(z0), EVEN''(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2'(true, z0) -> c18(COND1'(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2'(false, z0) -> c19(COND1'(neq(z0, 0), p(z0))) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, 0) -> c17(COND2'(true, 0), EVEN'(0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0)), EVEN'(s(0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) S tuples: COND1''(true, z0) -> c32(COND2''(even(z0), z0), EVEN'(z0), EVEN''(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) K tuples:none Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, COND1''_2, EVEN''_1, DIV2''_1, COND2'_2, COND2''_2, COND1'_2 Compound Symbols: c26_1, c29_1, c32_3, c43_1, c46_1, c18_2, c19_1, c33_2, c34_3, c35_1, c36_1, c17_2 ---------------------------------------- (13) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) COND1''(true, z0) -> c32(COND2''(even(z0), z0), EVEN'(z0), EVEN''(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2'(true, z0) -> c18(COND1'(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2'(false, z0) -> c19(COND1'(neq(z0, 0), p(z0))) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) S tuples: COND1''(true, z0) -> c32(COND2''(even(z0), z0), EVEN'(z0), EVEN''(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) K tuples:none Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, COND1''_2, EVEN''_1, DIV2''_1, COND2'_2, COND2''_2, COND1'_2 Compound Symbols: c26_1, c29_1, c32_3, c43_1, c46_1, c18_2, c19_1, c33_2, c34_3, c35_1, c36_1, c17_2, c17_1 ---------------------------------------- (15) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND1''(true, z0) -> c32(COND2''(even(z0), z0), EVEN'(z0), EVEN''(z0)) by COND1''(true, 0) -> c32(COND2''(true, 0), EVEN'(0), EVEN''(0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0)), EVEN'(s(0)), EVEN''(s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2'(true, z0) -> c18(COND1'(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2'(false, z0) -> c19(COND1'(neq(z0, 0), p(z0))) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0), EVEN'(0), EVEN''(0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0)), EVEN'(s(0)), EVEN''(s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, 0) -> c32(COND2''(true, 0), EVEN'(0), EVEN''(0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0)), EVEN'(s(0)), EVEN''(s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) K tuples:none Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2'_2, COND2''_2, COND1'_2, COND1''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c18_2, c19_1, c33_2, c34_3, c35_1, c36_1, c17_2, c17_1, c32_3 ---------------------------------------- (17) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2'(true, z0) -> c18(COND1'(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2'(false, z0) -> c19(COND1'(neq(z0, 0), p(z0))) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) K tuples:none Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2'_2, COND2''_2, COND1'_2, COND1''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c18_2, c19_1, c33_2, c34_3, c35_1, c36_1, c17_2, c17_1, c32_3, c32_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. COND1''(true, 0) -> c32(COND2''(true, 0)) We considered the (Usable) Rules: neq(s(z0), 0) -> true neq(0, 0) -> false And the Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2'(true, z0) -> c18(COND1'(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2'(false, z0) -> c19(COND1'(neq(z0, 0), p(z0))) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1'(x_1, x_2)) = 0 POL(COND1''(x_1, x_2)) = [1] + x_1 POL(COND2'(x_1, x_2)) = 0 POL(COND2''(x_1, x_2)) = [1] + x_2 POL(DIV2'(x_1)) = 0 POL(DIV2''(x_1)) = 0 POL(EVEN'(x_1)) = 0 POL(EVEN''(x_1)) = 0 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c32(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c33(x_1, x_2)) = x_1 + x_2 POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(div2(x_1)) = [1] + x_1 POL(even(x_1)) = [1] + x_1 POL(false) = 0 POL(neq(x_1, x_2)) = x_1 + x_2 POL(p(x_1)) = [1] + x_1 POL(s(x_1)) = [1] POL(true) = [1] ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2'(true, z0) -> c18(COND1'(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2'(false, z0) -> c19(COND1'(neq(z0, 0), p(z0))) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2'_2, COND2''_2, COND1'_2, COND1''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c18_2, c19_1, c33_2, c34_3, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1 ---------------------------------------- (21) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2'(true, z0) -> c18(COND1'(neq(z0, 0), div2(z0)), DIV2'(z0)) by COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0), DIV2'(0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0), DIV2'(s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, 0) -> c18(COND1'(false, div2(0)), DIV2'(0)) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2'(false, z0) -> c19(COND1'(neq(z0, 0), p(z0))) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0), DIV2'(0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0), DIV2'(s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, 0) -> c18(COND1'(false, div2(0)), DIV2'(0)) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2'_2, COND2''_2, COND1'_2, COND1''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c19_1, c33_2, c34_3, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: COND2'(true, 0) -> c18(COND1'(false, div2(0)), DIV2'(0)) ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2'(false, z0) -> c19(COND1'(neq(z0, 0), p(z0))) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0), DIV2'(0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0), DIV2'(s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2'_2, COND2''_2, COND1'_2, COND1''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c19_1, c33_2, c34_3, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2 ---------------------------------------- (25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2'(false, z0) -> c19(COND1'(neq(z0, 0), p(z0))) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2'_2, COND2''_2, COND1'_2, COND1''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c19_1, c33_2, c34_3, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1 ---------------------------------------- (27) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2'(false, z0) -> c19(COND1'(neq(z0, 0), p(z0))) by COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, 0) -> c19(COND1'(false, p(0))) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, 0) -> c19(COND1'(false, p(0))) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c33_2, c34_3, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1 ---------------------------------------- (29) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: COND2'(false, 0) -> c19(COND1'(false, p(0))) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c33_2, c34_3, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1 ---------------------------------------- (31) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2''(true, z0) -> c33(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0)) by COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0), DIV2'(0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0), DIV2'(s(0))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c33(COND1''(false, div2(0)), DIV2'(0)) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0), DIV2'(0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0), DIV2'(s(0))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c33(COND1''(false, div2(0)), DIV2'(0)) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0), DIV2'(0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0), DIV2'(s(0))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c33(COND1''(false, div2(0)), DIV2'(0)) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c34_3, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2 ---------------------------------------- (33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: COND2''(true, 0) -> c33(COND1''(false, div2(0)), DIV2'(0)) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0), DIV2'(0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0), DIV2'(s(0))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0), DIV2'(0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0), DIV2'(s(0))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c34_3, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2 ---------------------------------------- (35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c34_3, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1 ---------------------------------------- (37) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, 0) -> c32(COND2''(true, 0)) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c34_3, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1 ---------------------------------------- (39) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) We considered the (Usable) Rules: div2(0) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 div2(s(0)) -> 0 And the Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1'(x_1, x_2)) = 0 POL(COND1''(x_1, x_2)) = [1] + x_2 POL(COND2'(x_1, x_2)) = 0 POL(COND2''(x_1, x_2)) = [1] + x_2 POL(DIV2'(x_1)) = 0 POL(DIV2''(x_1)) = 0 POL(EVEN'(x_1)) = 0 POL(EVEN''(x_1)) = 0 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c32(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c33(x_1)) = x_1 POL(c33(x_1, x_2)) = x_1 + x_2 POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(div2(x_1)) = x_1 POL(even(x_1)) = 0 POL(false) = 0 POL(neq(x_1, x_2)) = 0 POL(p(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c34_3, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1 ---------------------------------------- (41) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2''(true, z0) -> c34(COND1''(neq(z0, 0), div2(z0)), DIV2'(z0), DIV2''(z0)) by COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0), DIV2'(0), DIV2''(0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0), DIV2'(s(0)), DIV2''(s(0))) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(false, div2(0)), DIV2'(0), DIV2''(0)) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0), DIV2'(0), DIV2''(0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0), DIV2'(s(0)), DIV2''(s(0))) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(false, div2(0)), DIV2'(0), DIV2''(0)) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0), DIV2'(0), DIV2''(0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0), DIV2'(s(0)), DIV2''(s(0))) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(false, div2(0)), DIV2'(0), DIV2''(0)) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3 ---------------------------------------- (43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: COND2''(true, 0) -> c34(COND1''(false, div2(0)), DIV2'(0), DIV2''(0)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0), DIV2'(0), DIV2''(0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0), DIV2'(s(0)), DIV2''(s(0))) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0), DIV2'(0), DIV2''(0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0), DIV2'(s(0)), DIV2''(s(0))) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3 ---------------------------------------- (45) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1 ---------------------------------------- (47) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, 0) -> c32(COND2''(true, 0)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1 ---------------------------------------- (49) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) We considered the (Usable) Rules: div2(0) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 div2(s(0)) -> 0 And the Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1'(x_1, x_2)) = 0 POL(COND1''(x_1, x_2)) = x_2 POL(COND2'(x_1, x_2)) = 0 POL(COND2''(x_1, x_2)) = x_2 POL(DIV2'(x_1)) = 0 POL(DIV2''(x_1)) = 0 POL(EVEN'(x_1)) = 0 POL(EVEN''(x_1)) = 0 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c32(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c33(x_1)) = x_1 POL(c33(x_1, x_2)) = x_1 + x_2 POL(c34(x_1)) = x_1 POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(div2(x_1)) = x_1 POL(even(x_1)) = [1] + x_1 POL(false) = 0 POL(neq(x_1, x_2)) = 0 POL(p(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c35_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2''(false, z0) -> c35(COND1''(neq(z0, 0), p(z0))) by COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c35(COND1''(false, p(0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c35(COND1''(false, p(0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c35(COND1''(false, p(0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1 ---------------------------------------- (53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: COND2''(false, 0) -> c35(COND1''(false, p(0))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1 ---------------------------------------- (55) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND1''(true, 0) -> c32(COND2''(true, 0)) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1 ---------------------------------------- (57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) We considered the (Usable) Rules: div2(0) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 div2(s(0)) -> 0 And the Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1'(x_1, x_2)) = 0 POL(COND1''(x_1, x_2)) = x_2 POL(COND2'(x_1, x_2)) = 0 POL(COND2''(x_1, x_2)) = x_2 POL(DIV2'(x_1)) = 0 POL(DIV2''(x_1)) = 0 POL(EVEN'(x_1)) = 0 POL(EVEN''(x_1)) = 0 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c32(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c33(x_1)) = x_1 POL(c33(x_1, x_2)) = x_1 + x_2 POL(c34(x_1)) = x_1 POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(div2(x_1)) = x_1 POL(even(x_1)) = [1] + x_1 POL(false) = 0 POL(neq(x_1, x_2)) = 0 POL(p(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND2''_2, COND1'_2, COND1''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c36_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2''(false, z0) -> c36(COND1''(neq(z0, 0), p(z0))) by COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(false, p(0))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(false, p(0))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(false, p(0))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1 ---------------------------------------- (61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: COND2''(false, 0) -> c36(COND1''(false, p(0))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1 ---------------------------------------- (63) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND1''(true, 0) -> c32(COND2''(true, 0)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(0, 0) -> false neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(0) -> 0 p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1 ---------------------------------------- (65) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: p(0) -> 0 ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1 ---------------------------------------- (67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) We considered the (Usable) Rules: div2(0) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 div2(s(0)) -> 0 And the Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1'(x_1, x_2)) = 0 POL(COND1''(x_1, x_2)) = x_2 POL(COND2'(x_1, x_2)) = 0 POL(COND2''(x_1, x_2)) = x_2 POL(DIV2'(x_1)) = 0 POL(DIV2''(x_1)) = 0 POL(EVEN'(x_1)) = 0 POL(EVEN''(x_1)) = 0 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c32(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c33(x_1)) = x_1 POL(c33(x_1, x_2)) = x_1 + x_2 POL(c34(x_1)) = x_1 POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(div2(x_1)) = x_1 POL(even(x_1)) = [1] + x_1 POL(false) = 0 POL(neq(x_1, x_2)) = 0 POL(p(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_2, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND1'(true, s(s(z0))) -> c17(COND2'(even(z0), s(s(z0))), EVEN'(s(s(z0)))) by COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_3, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND1''(true, s(s(z0))) -> c32(COND2''(even(z0), s(s(z0))), EVEN'(s(s(z0))), EVEN''(s(s(z0)))) by COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c32(EVEN'(s(s(x0))), EVEN''(s(s(x0)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c32(EVEN'(s(s(x0))), EVEN''(s(s(x0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c32(EVEN'(s(s(x0))), EVEN''(s(s(x0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c32_2 ---------------------------------------- (73) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1 ---------------------------------------- (75) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) We considered the (Usable) Rules:none And the Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1'(x_1, x_2)) = 0 POL(COND1''(x_1, x_2)) = [1] POL(COND2'(x_1, x_2)) = x_2 POL(COND2''(x_1, x_2)) = [1] + x_2 POL(DIV2'(x_1)) = 0 POL(DIV2''(x_1)) = 0 POL(EVEN'(x_1)) = 0 POL(EVEN''(x_1)) = 0 POL(c(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c32(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c33(x_1)) = x_1 POL(c33(x_1, x_2)) = x_1 + x_2 POL(c34(x_1)) = x_1 POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(div2(x_1)) = 0 POL(even(x_1)) = [1] + x_1 POL(false) = 0 POL(neq(x_1, x_2)) = 0 POL(p(x_1)) = [1] POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2'(true, s(s(z0))) -> c18(COND1'(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) by COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(0)))) -> c18(COND1'(neq(s(s(s(0))), 0), s(0)), DIV2'(s(s(s(0))))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(0)))) -> c18(COND1'(neq(s(s(s(0))), 0), s(0)), DIV2'(s(s(s(0))))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1 ---------------------------------------- (79) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(true, s(s(s(0)))) -> c1(DIV2'(s(s(s(0))))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c1_1 ---------------------------------------- (81) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: COND2'(true, s(s(s(0)))) -> c1(DIV2'(s(s(s(0))))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c18_2, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c1_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2'(true, s(z0)) -> c18(COND1'(true, div2(s(z0))), DIV2'(s(z0))) by COND2'(true, s(0)) -> c18(COND1'(true, 0), DIV2'(s(0))) COND2'(true, s(s(z0))) -> c18(COND1'(true, s(div2(z0))), DIV2'(s(s(z0)))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(true, s(0)) -> c18(COND1'(true, 0), DIV2'(s(0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c1_1 ---------------------------------------- (85) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: COND2'(true, s(0)) -> c18(COND1'(true, 0), DIV2'(s(0))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c1_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2'(true, 0) -> c18(COND1'(neq(0, 0), 0)) by COND2'(true, 0) -> c18(COND1'(false, 0)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, 0) -> c17(COND2'(true, 0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(true, 0) -> c18(COND1'(false, 0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c18_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c1_1 ---------------------------------------- (89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: COND1'(true, 0) -> c17(COND2'(true, 0)) COND2'(true, s(0)) -> c18(COND1'(neq(s(0), 0), 0)) COND2'(true, 0) -> c18(COND1'(false, 0)) COND2'(false, 0) -> c19(COND1'(neq(0, 0), 0)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2'(false, s(z0)) -> c19(COND1'(neq(s(z0), 0), z0)) by COND2'(false, s(z0)) -> c19(COND1'(true, z0)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c19_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2'(false, s(z0)) -> c19(COND1'(true, p(s(z0)))) by COND2'(false, s(z0)) -> c19(COND1'(true, z0)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) by COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(0)))) -> c33(COND1''(neq(s(s(s(0))), 0), s(0)), DIV2'(s(s(s(0))))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(0)))) -> c33(COND1''(neq(s(s(s(0))), 0), s(0)), DIV2'(s(s(s(0))))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1 ---------------------------------------- (97) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(s(0)))) -> c2(DIV2'(s(s(s(0))))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c33(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0)))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c2_1 ---------------------------------------- (99) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: COND2''(true, s(s(s(0)))) -> c2(DIV2'(s(s(s(0))))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c33_2, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c2_1 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2''(true, s(z0)) -> c33(COND1''(true, div2(s(z0))), DIV2'(s(z0))) by COND2''(true, s(0)) -> c33(COND1''(true, 0), DIV2'(s(0))) COND2''(true, s(s(z0))) -> c33(COND1''(true, s(div2(z0))), DIV2'(s(s(z0)))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(0)) -> c33(COND1''(true, 0), DIV2'(s(0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND2''(true, s(0)) -> c33(COND1''(true, 0), DIV2'(s(0))) COND2''(true, s(s(z0))) -> c33(COND1''(true, s(div2(z0))), DIV2'(s(s(z0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c2_1 ---------------------------------------- (103) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: COND2''(true, s(0)) -> c33(COND1''(true, 0), DIV2'(s(0))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND2''(true, s(s(z0))) -> c33(COND1''(true, s(div2(z0))), DIV2'(s(s(z0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c2_1 ---------------------------------------- (105) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) We considered the (Usable) Rules: div2(0) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 div2(s(0)) -> 0 And the Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1'(x_1, x_2)) = 0 POL(COND1''(x_1, x_2)) = x_2 POL(COND2'(x_1, x_2)) = 0 POL(COND2''(x_1, x_2)) = x_2 POL(DIV2'(x_1)) = 0 POL(DIV2''(x_1)) = 0 POL(EVEN'(x_1)) = 0 POL(EVEN''(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c32(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c33(x_1)) = x_1 POL(c33(x_1, x_2)) = x_1 + x_2 POL(c34(x_1)) = x_1 POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(div2(x_1)) = x_1 POL(even(x_1)) = [1] + x_1 POL(false) = 0 POL(neq(x_1, x_2)) = 0 POL(p(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c2_1 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) by COND2''(true, 0) -> c33(COND1''(false, 0)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, 0) -> c33(COND1''(false, 0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c33(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c2_1 ---------------------------------------- (109) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: COND2''(true, 0) -> c33(COND1''(false, 0)) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c33_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c2_1 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) by COND2''(true, s(0)) -> c33(COND1''(true, 0)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(0)) -> c33(COND1''(true, 0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, s(0)) -> c33(COND1''(neq(s(0), 0), 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1 ---------------------------------------- (113) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: COND2''(true, s(0)) -> c33(COND1''(true, 0)) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) by COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(0)))) -> c34(COND1''(neq(s(s(s(0))), 0), s(0)), DIV2'(s(s(s(0)))), DIV2''(s(s(s(0))))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(DIV2'(s(s(x0))), DIV2''(s(s(x0)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(0)))) -> c34(COND1''(neq(s(s(s(0))), 0), s(0)), DIV2'(s(s(s(0)))), DIV2''(s(s(s(0))))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(DIV2'(s(s(x0))), DIV2''(s(s(x0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_2 ---------------------------------------- (117) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(s(0)))) -> c3(DIV2'(s(s(s(0))))) COND2''(true, s(s(s(0)))) -> c3(DIV2''(s(s(s(0))))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(true, s(s(z0))) -> c34(COND1''(neq(s(s(z0)), 0), s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c3_1 ---------------------------------------- (119) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: COND2''(true, s(s(s(0)))) -> c3(DIV2'(s(s(s(0))))) COND2''(true, s(s(s(0)))) -> c3(DIV2''(s(s(s(0))))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c34_3, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c3_1 ---------------------------------------- (121) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2''(true, s(z0)) -> c34(COND1''(true, div2(s(z0))), DIV2'(s(z0)), DIV2''(s(z0))) by COND2''(true, s(0)) -> c34(COND1''(true, 0), DIV2'(s(0)), DIV2''(s(0))) COND2''(true, s(s(z0))) -> c34(COND1''(true, s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) COND2''(true, s(0)) -> c34(COND1''(true, 0), DIV2'(s(0)), DIV2''(s(0))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND2''(true, s(0)) -> c34(COND1''(true, 0), DIV2'(s(0)), DIV2''(s(0))) COND2''(true, s(s(z0))) -> c34(COND1''(true, s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1 ---------------------------------------- (123) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: COND2''(true, s(0)) -> c34(COND1''(true, 0), DIV2'(s(0)), DIV2''(s(0))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND2''(true, s(s(z0))) -> c34(COND1''(true, s(div2(z0))), DIV2'(s(s(z0))), DIV2''(s(s(z0)))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1 ---------------------------------------- (125) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) We considered the (Usable) Rules: div2(0) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 div2(s(0)) -> 0 And the Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1'(x_1, x_2)) = 0 POL(COND1''(x_1, x_2)) = x_2 POL(COND2'(x_1, x_2)) = 0 POL(COND2''(x_1, x_2)) = x_2 POL(DIV2'(x_1)) = 0 POL(DIV2''(x_1)) = 0 POL(EVEN'(x_1)) = 0 POL(EVEN''(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c32(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c33(x_1)) = x_1 POL(c33(x_1, x_2)) = x_1 + x_2 POL(c34(x_1)) = x_1 POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(div2(x_1)) = x_1 POL(even(x_1)) = [1] + x_1 POL(false) = 0 POL(neq(x_1, x_2)) = x_2 POL(p(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) by COND2''(true, 0) -> c34(COND1''(false, 0)) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, 0) -> c32(COND2''(true, 0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) COND2''(true, 0) -> c34(COND1''(false, 0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c34(COND1''(neq(0, 0), 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c34_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1 ---------------------------------------- (129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing nodes: COND2''(false, 0) -> c36(COND1''(neq(0, 0), 0)) COND2''(false, 0) -> c35(COND1''(neq(0, 0), 0)) COND1''(true, 0) -> c32(COND2''(true, 0)) COND2''(true, 0) -> c34(COND1''(false, 0)) COND2''(true, s(0)) -> c34(COND1''(neq(s(0), 0), 0)) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true neq(0, 0) -> false div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) Defined Rule Symbols: even_1, neq_2, div2_1, p_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1 ---------------------------------------- (131) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: neq(0, 0) -> false ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: neq(s(z0), 0) -> true p(s(z0)) -> z0 even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) Defined Rule Symbols: neq_2, p_1, even_1, div2_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1 ---------------------------------------- (133) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) We considered the (Usable) Rules: div2(0) -> 0 div2(s(s(z0))) -> s(div2(z0)) even(0) -> true p(s(z0)) -> z0 even(s(0)) -> false div2(s(0)) -> 0 even(s(s(z0))) -> even(z0) And the Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1'(x_1, x_2)) = 0 POL(COND1''(x_1, x_2)) = [1] + x_2 POL(COND2'(x_1, x_2)) = 0 POL(COND2''(x_1, x_2)) = x_1 + x_2 POL(DIV2'(x_1)) = 0 POL(DIV2''(x_1)) = 0 POL(EVEN'(x_1)) = 0 POL(EVEN''(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c32(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c33(x_1)) = x_1 POL(c33(x_1, x_2)) = x_1 + x_2 POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(div2(x_1)) = x_1 POL(even(x_1)) = [1] POL(false) = [1] POL(neq(x_1, x_2)) = [1] + x_2 POL(p(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: neq(s(z0), 0) -> true p(s(z0)) -> z0 even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) Defined Rule Symbols: neq_2, p_1, even_1, div2_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1 ---------------------------------------- (135) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. DIV2''(s(s(z0))) -> c46(DIV2''(z0)) We considered the (Usable) Rules: div2(0) -> 0 div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 div2(s(0)) -> 0 And the Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1'(x_1, x_2)) = x_2 POL(COND1''(x_1, x_2)) = x_2^2 POL(COND2'(x_1, x_2)) = x_2 POL(COND2''(x_1, x_2)) = x_2^2 POL(DIV2'(x_1)) = [2] POL(DIV2''(x_1)) = [2] + x_1 POL(EVEN'(x_1)) = 0 POL(EVEN''(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c32(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c33(x_1)) = x_1 POL(c33(x_1, x_2)) = x_1 + x_2 POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(div2(x_1)) = x_1 POL(even(x_1)) = 0 POL(false) = 0 POL(neq(x_1, x_2)) = 0 POL(p(x_1)) = x_1 POL(s(x_1)) = [2] + x_1 POL(true) = 0 ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: neq(s(z0), 0) -> true p(s(z0)) -> z0 even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) Defined Rule Symbols: neq_2, p_1, even_1, div2_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) by COND2''(false, s(z0)) -> c35(COND1''(true, z0)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: neq(s(z0), 0) -> true p(s(z0)) -> z0 even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) Defined Rule Symbols: neq_2, p_1, even_1, div2_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c35_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2''(false, s(z0)) -> c35(COND1''(true, p(s(z0)))) by COND2''(false, s(z0)) -> c35(COND1''(true, z0)) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: neq(s(z0), 0) -> true p(s(z0)) -> z0 even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) K tuples: COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) Defined Rule Symbols: neq_2, p_1, even_1, div2_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1, c35_1 ---------------------------------------- (141) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND2''(false, s(z0)) -> c35(COND1''(true, z0)) We considered the (Usable) Rules: div2(0) -> 0 neq(s(z0), 0) -> true div2(s(s(z0))) -> s(div2(z0)) p(s(z0)) -> z0 div2(s(0)) -> 0 And the Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1'(x_1, x_2)) = 0 POL(COND1''(x_1, x_2)) = x_1 + x_2 POL(COND2'(x_1, x_2)) = 0 POL(COND2''(x_1, x_2)) = x_2 POL(DIV2'(x_1)) = 0 POL(DIV2''(x_1)) = 0 POL(EVEN'(x_1)) = 0 POL(EVEN''(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c32(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c33(x_1)) = x_1 POL(c33(x_1, x_2)) = x_1 + x_2 POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(div2(x_1)) = [1] + x_1 POL(even(x_1)) = [1] POL(false) = [1] POL(neq(x_1, x_2)) = [1] + x_2 POL(p(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: neq(s(z0), 0) -> true p(s(z0)) -> z0 even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) Defined Rule Symbols: neq_2, p_1, even_1, div2_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1, c35_1 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) by COND2''(false, s(z0)) -> c36(COND1''(true, z0)) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: neq(s(z0), 0) -> true p(s(z0)) -> z0 even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) COND2''(false, s(z0)) -> c36(COND1''(true, z0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) Defined Rule Symbols: neq_2, p_1, even_1, div2_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2''_2, COND2'_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c36_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1, c35_1 ---------------------------------------- (145) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND2''(false, s(z0)) -> c36(COND1''(true, p(s(z0)))) by COND2''(false, s(z0)) -> c36(COND1''(true, z0)) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: neq(s(z0), 0) -> true p(s(z0)) -> z0 even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) COND2''(false, s(z0)) -> c36(COND1''(true, z0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND2''(false, s(z0)) -> c36(COND1''(true, z0)) K tuples: COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) Defined Rule Symbols: neq_2, p_1, even_1, div2_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1, c35_1, c36_1 ---------------------------------------- (147) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: p(s(z0)) -> z0 ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) COND2''(false, s(z0)) -> c36(COND1''(true, z0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND2''(false, s(z0)) -> c36(COND1''(true, z0)) K tuples: COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) Defined Rule Symbols: even_1, neq_2, div2_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1, c35_1, c36_1 ---------------------------------------- (149) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND2''(false, s(z0)) -> c36(COND1''(true, z0)) We considered the (Usable) Rules: div2(0) -> 0 neq(s(z0), 0) -> true div2(s(s(z0))) -> s(div2(z0)) even(0) -> true even(s(0)) -> false div2(s(0)) -> 0 even(s(s(z0))) -> even(z0) And the Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) COND2''(false, s(z0)) -> c36(COND1''(true, z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1'(x_1, x_2)) = x_2 POL(COND1''(x_1, x_2)) = [1] + x_1 + x_2 POL(COND2'(x_1, x_2)) = x_2 POL(COND2''(x_1, x_2)) = [1] + x_1 + x_2 POL(DIV2'(x_1)) = [1] POL(DIV2''(x_1)) = 0 POL(EVEN'(x_1)) = 0 POL(EVEN''(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c32(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c33(x_1)) = x_1 POL(c33(x_1, x_2)) = x_1 + x_2 POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(div2(x_1)) = x_1 POL(even(x_1)) = [1] POL(false) = [1] POL(neq(x_1, x_2)) = [1] + x_2 POL(s(x_1)) = [1] + x_1 POL(true) = [1] ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) COND2''(false, s(z0)) -> c36(COND1''(true, z0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) K tuples: COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) COND2''(false, s(z0)) -> c36(COND1''(true, z0)) Defined Rule Symbols: even_1, neq_2, div2_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1, c35_1, c36_1 ---------------------------------------- (151) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) We considered the (Usable) Rules: div2(0) -> 0 neq(s(z0), 0) -> true div2(s(s(z0))) -> s(div2(z0)) div2(s(0)) -> 0 And the Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) COND2''(false, s(z0)) -> c36(COND1''(true, z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1'(x_1, x_2)) = 0 POL(COND1''(x_1, x_2)) = x_1 + x_2 POL(COND2'(x_1, x_2)) = 0 POL(COND2''(x_1, x_2)) = x_2 POL(DIV2'(x_1)) = 0 POL(DIV2''(x_1)) = 0 POL(EVEN'(x_1)) = 0 POL(EVEN''(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c32(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c33(x_1)) = x_1 POL(c33(x_1, x_2)) = x_1 + x_2 POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(div2(x_1)) = x_1 POL(even(x_1)) = [1] + x_1 POL(false) = [1] POL(neq(x_1, x_2)) = [1] + x_2 POL(s(x_1)) = [1] + x_1 POL(true) = [1] ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) COND2''(false, s(z0)) -> c36(COND1''(true, z0)) S tuples: EVEN''(s(s(z0))) -> c43(EVEN''(z0)) K tuples: COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) COND2''(false, s(z0)) -> c36(COND1''(true, z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) Defined Rule Symbols: even_1, neq_2, div2_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1, c35_1, c36_1 ---------------------------------------- (153) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. EVEN''(s(s(z0))) -> c43(EVEN''(z0)) We considered the (Usable) Rules: div2(0) -> 0 neq(s(z0), 0) -> true div2(s(s(z0))) -> s(div2(z0)) div2(s(0)) -> 0 And the Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) COND2''(false, s(z0)) -> c36(COND1''(true, z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND1'(x_1, x_2)) = 0 POL(COND1''(x_1, x_2)) = x_2^2 + [2]x_1*x_2 POL(COND2'(x_1, x_2)) = 0 POL(COND2''(x_1, x_2)) = [1] + x_2^2 POL(DIV2'(x_1)) = 0 POL(DIV2''(x_1)) = 0 POL(EVEN'(x_1)) = 0 POL(EVEN''(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c18(x_1, x_2)) = x_1 + x_2 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c32(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c33(x_1)) = x_1 POL(c33(x_1, x_2)) = x_1 + x_2 POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(div2(x_1)) = x_1 POL(even(x_1)) = 0 POL(false) = 0 POL(neq(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = [1] ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) neq(s(z0), 0) -> true div2(0) -> 0 div2(s(0)) -> 0 div2(s(s(z0))) -> s(div2(z0)) Tuples: EVEN'(s(s(z0))) -> c26(EVEN'(z0)) DIV2'(s(s(z0))) -> c29(DIV2'(z0)) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND1'(true, s(0)) -> c17(COND2'(false, s(0))) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND1'(true, s(s(0))) -> c17(COND2'(true, s(s(0))), EVEN'(s(s(0)))) COND1'(true, s(s(s(0)))) -> c17(COND2'(false, s(s(s(0)))), EVEN'(s(s(s(0))))) COND1'(true, s(s(s(s(z0))))) -> c17(COND2'(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0)))))) COND1'(true, s(s(x0))) -> c17(EVEN'(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2'(true, s(s(0))) -> c18(COND1'(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2'(true, s(s(s(s(z0))))) -> c18(COND1'(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2'(true, s(s(x0))) -> c18(COND1'(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2'(true, s(s(x0))) -> c18(DIV2'(s(s(x0)))) COND2'(true, s(s(s(0)))) -> c1(COND1'(neq(s(s(s(0))), 0), s(0))) COND2'(false, s(z0)) -> c19(COND1'(true, z0)) COND2''(true, s(s(0))) -> c33(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c33(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(DIV2'(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c2(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(0))) -> c34(COND1''(neq(s(s(0)), 0), s(0)), DIV2'(s(s(0))), DIV2''(s(s(0)))) COND2''(true, s(s(s(s(z0))))) -> c34(COND1''(neq(s(s(s(s(z0)))), 0), s(s(div2(z0)))), DIV2'(s(s(s(s(z0))))), DIV2''(s(s(s(s(z0)))))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND2''(true, s(s(s(0)))) -> c3(COND1''(neq(s(s(s(0))), 0), s(0))) COND2''(true, s(s(x0))) -> c3(DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c3(DIV2''(s(s(x0)))) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) COND2''(false, s(z0)) -> c36(COND1''(true, z0)) S tuples:none K tuples: COND2''(false, s(z0)) -> c35(COND1''(neq(s(z0), 0), z0)) COND2''(false, s(z0)) -> c36(COND1''(neq(s(z0), 0), z0)) COND1''(true, s(s(x0))) -> c(EVEN'(s(s(x0)))) COND1''(true, s(s(x0))) -> c(EVEN''(s(s(x0)))) COND2''(true, s(s(x0))) -> c33(COND1''(true, s(div2(x0))), DIV2'(s(s(x0)))) COND2''(true, s(s(x0))) -> c34(COND1''(true, s(div2(x0))), DIV2'(s(s(x0))), DIV2''(s(s(x0)))) COND1''(true, s(s(0))) -> c32(COND2''(true, s(s(0))), EVEN'(s(s(0))), EVEN''(s(s(0)))) DIV2''(s(s(z0))) -> c46(DIV2''(z0)) COND2''(false, s(z0)) -> c35(COND1''(true, z0)) COND2''(false, s(z0)) -> c36(COND1''(true, z0)) COND1''(true, s(0)) -> c32(COND2''(false, s(0))) COND1''(true, s(s(s(0)))) -> c32(COND2''(false, s(s(s(0)))), EVEN'(s(s(s(0)))), EVEN''(s(s(s(0))))) COND1''(true, s(s(s(s(z0))))) -> c32(COND2''(even(z0), s(s(s(s(z0))))), EVEN'(s(s(s(s(z0))))), EVEN''(s(s(s(s(z0)))))) EVEN''(s(s(z0))) -> c43(EVEN''(z0)) Defined Rule Symbols: even_1, neq_2, div2_1 Defined Pair Symbols: EVEN'_1, DIV2'_1, EVEN''_1, DIV2''_1, COND1'_2, COND1''_2, COND2'_2, COND2''_2 Compound Symbols: c26_1, c29_1, c43_1, c46_1, c17_1, c32_1, c17_2, c32_3, c_1, c18_2, c18_1, c1_1, c19_1, c33_2, c33_1, c2_1, c34_3, c3_1, c35_1, c36_1 ---------------------------------------- (155) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (156) BOUNDS(1, 1) ---------------------------------------- (157) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (158) 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: COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0)) COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0')) COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0)) COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0')) COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0)) NEQ(0', 0') -> c5 NEQ(0', s(z0)) -> c6 NEQ(s(z0), 0') -> c7 NEQ(s(z0), s(y)) -> c8(NEQ(z0, y)) EVEN(0') -> c9 EVEN(s(0')) -> c10 EVEN(s(s(z0))) -> c11(EVEN(z0)) DIV2(0') -> c12 DIV2(s(0')) -> c13 DIV2(s(s(z0))) -> c14(DIV2(z0)) P(0') -> c15 P(s(z0)) -> c16 The (relative) TRS S consists of the following rules: cond1(true, z0) -> cond2(even(z0), z0) cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0)) cond2(false, z0) -> cond1(neq(z0, 0'), p(z0)) neq(0', 0') -> false neq(0', s(z0)) -> true neq(s(z0), 0') -> true neq(s(z0), s(y)) -> neq(z0, y) even(0') -> true even(s(0')) -> false even(s(s(z0))) -> even(z0) div2(0') -> 0' div2(s(0')) -> 0' div2(s(s(z0))) -> s(div2(z0)) p(0') -> 0' p(s(z0)) -> z0 Rewrite Strategy: INNERMOST ---------------------------------------- (159) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (160) Obligation: Innermost TRS: Rules: COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0)) COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0')) COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0)) COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0')) COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0)) NEQ(0', 0') -> c5 NEQ(0', s(z0)) -> c6 NEQ(s(z0), 0') -> c7 NEQ(s(z0), s(y)) -> c8(NEQ(z0, y)) EVEN(0') -> c9 EVEN(s(0')) -> c10 EVEN(s(s(z0))) -> c11(EVEN(z0)) DIV2(0') -> c12 DIV2(s(0')) -> c13 DIV2(s(s(z0))) -> c14(DIV2(z0)) P(0') -> c15 P(s(z0)) -> c16 cond1(true, z0) -> cond2(even(z0), z0) cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0)) cond2(false, z0) -> cond1(neq(z0, 0'), p(z0)) neq(0', 0') -> false neq(0', s(z0)) -> true neq(s(z0), 0') -> true neq(s(z0), s(y)) -> neq(z0, y) even(0') -> true even(s(0')) -> false even(s(s(z0))) -> even(z0) div2(0') -> 0' div2(s(0')) -> 0' div2(s(s(z0))) -> s(div2(z0)) p(0') -> 0' p(s(z0)) -> z0 Types: COND1 :: true:false -> 0':s:y -> c true :: true:false c :: c1:c2:c3:c4 -> c9:c10:c11 -> c COND2 :: true:false -> 0':s:y -> c1:c2:c3:c4 even :: 0':s:y -> true:false EVEN :: 0':s:y -> c9:c10:c11 c1 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4 neq :: 0':s:y -> 0':s:y -> true:false 0' :: 0':s:y div2 :: 0':s:y -> 0':s:y NEQ :: 0':s:y -> 0':s:y -> c5:c6:c7:c8 c2 :: c -> c12:c13:c14 -> c1:c2:c3:c4 DIV2 :: 0':s:y -> c12:c13:c14 false :: true:false c3 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4 p :: 0':s:y -> 0':s:y c4 :: c -> c15:c16 -> c1:c2:c3:c4 P :: 0':s:y -> c15:c16 c5 :: c5:c6:c7:c8 s :: 0':s:y -> 0':s:y c6 :: c5:c6:c7:c8 c7 :: c5:c6:c7:c8 y :: 0':s:y c8 :: c5:c6:c7:c8 -> c5:c6:c7:c8 c9 :: c9:c10:c11 c10 :: c9:c10:c11 c11 :: c9:c10:c11 -> c9:c10:c11 c12 :: c12:c13:c14 c13 :: c12:c13:c14 c14 :: c12:c13:c14 -> c12:c13:c14 c15 :: c15:c16 c16 :: c15:c16 cond1 :: true:false -> 0':s:y -> cond1:cond2 cond2 :: true:false -> 0':s:y -> cond1:cond2 hole_c1_17 :: c hole_true:false2_17 :: true:false hole_0':s:y3_17 :: 0':s:y hole_c1:c2:c3:c44_17 :: c1:c2:c3:c4 hole_c9:c10:c115_17 :: c9:c10:c11 hole_c5:c6:c7:c86_17 :: c5:c6:c7:c8 hole_c12:c13:c147_17 :: c12:c13:c14 hole_c15:c168_17 :: c15:c16 hole_cond1:cond29_17 :: cond1:cond2 gen_0':s:y10_17 :: Nat -> 0':s:y gen_c9:c10:c1111_17 :: Nat -> c9:c10:c11 gen_c5:c6:c7:c812_17 :: Nat -> c5:c6:c7:c8 gen_c12:c13:c1413_17 :: Nat -> c12:c13:c14 ---------------------------------------- (161) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: COND1, COND2, even, EVEN, neq, div2, NEQ, DIV2, cond1, cond2 They will be analysed ascendingly in the following order: COND1 = COND2 even < COND1 EVEN < COND1 neq < COND2 div2 < COND2 NEQ < COND2 DIV2 < COND2 even < cond1 neq < cond2 div2 < cond2 cond1 = cond2 ---------------------------------------- (162) Obligation: Innermost TRS: Rules: COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0)) COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0')) COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0)) COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0')) COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0)) NEQ(0', 0') -> c5 NEQ(0', s(z0)) -> c6 NEQ(s(z0), 0') -> c7 NEQ(s(z0), s(y)) -> c8(NEQ(z0, y)) EVEN(0') -> c9 EVEN(s(0')) -> c10 EVEN(s(s(z0))) -> c11(EVEN(z0)) DIV2(0') -> c12 DIV2(s(0')) -> c13 DIV2(s(s(z0))) -> c14(DIV2(z0)) P(0') -> c15 P(s(z0)) -> c16 cond1(true, z0) -> cond2(even(z0), z0) cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0)) cond2(false, z0) -> cond1(neq(z0, 0'), p(z0)) neq(0', 0') -> false neq(0', s(z0)) -> true neq(s(z0), 0') -> true neq(s(z0), s(y)) -> neq(z0, y) even(0') -> true even(s(0')) -> false even(s(s(z0))) -> even(z0) div2(0') -> 0' div2(s(0')) -> 0' div2(s(s(z0))) -> s(div2(z0)) p(0') -> 0' p(s(z0)) -> z0 Types: COND1 :: true:false -> 0':s:y -> c true :: true:false c :: c1:c2:c3:c4 -> c9:c10:c11 -> c COND2 :: true:false -> 0':s:y -> c1:c2:c3:c4 even :: 0':s:y -> true:false EVEN :: 0':s:y -> c9:c10:c11 c1 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4 neq :: 0':s:y -> 0':s:y -> true:false 0' :: 0':s:y div2 :: 0':s:y -> 0':s:y NEQ :: 0':s:y -> 0':s:y -> c5:c6:c7:c8 c2 :: c -> c12:c13:c14 -> c1:c2:c3:c4 DIV2 :: 0':s:y -> c12:c13:c14 false :: true:false c3 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4 p :: 0':s:y -> 0':s:y c4 :: c -> c15:c16 -> c1:c2:c3:c4 P :: 0':s:y -> c15:c16 c5 :: c5:c6:c7:c8 s :: 0':s:y -> 0':s:y c6 :: c5:c6:c7:c8 c7 :: c5:c6:c7:c8 y :: 0':s:y c8 :: c5:c6:c7:c8 -> c5:c6:c7:c8 c9 :: c9:c10:c11 c10 :: c9:c10:c11 c11 :: c9:c10:c11 -> c9:c10:c11 c12 :: c12:c13:c14 c13 :: c12:c13:c14 c14 :: c12:c13:c14 -> c12:c13:c14 c15 :: c15:c16 c16 :: c15:c16 cond1 :: true:false -> 0':s:y -> cond1:cond2 cond2 :: true:false -> 0':s:y -> cond1:cond2 hole_c1_17 :: c hole_true:false2_17 :: true:false hole_0':s:y3_17 :: 0':s:y hole_c1:c2:c3:c44_17 :: c1:c2:c3:c4 hole_c9:c10:c115_17 :: c9:c10:c11 hole_c5:c6:c7:c86_17 :: c5:c6:c7:c8 hole_c12:c13:c147_17 :: c12:c13:c14 hole_c15:c168_17 :: c15:c16 hole_cond1:cond29_17 :: cond1:cond2 gen_0':s:y10_17 :: Nat -> 0':s:y gen_c9:c10:c1111_17 :: Nat -> c9:c10:c11 gen_c5:c6:c7:c812_17 :: Nat -> c5:c6:c7:c8 gen_c12:c13:c1413_17 :: Nat -> c12:c13:c14 Generator Equations: gen_0':s:y10_17(0) <=> 0' gen_0':s:y10_17(+(x, 1)) <=> s(gen_0':s:y10_17(x)) gen_c9:c10:c1111_17(0) <=> c9 gen_c9:c10:c1111_17(+(x, 1)) <=> c11(gen_c9:c10:c1111_17(x)) gen_c5:c6:c7:c812_17(0) <=> c5 gen_c5:c6:c7:c812_17(+(x, 1)) <=> c8(gen_c5:c6:c7:c812_17(x)) gen_c12:c13:c1413_17(0) <=> c12 gen_c12:c13:c1413_17(+(x, 1)) <=> c14(gen_c12:c13:c1413_17(x)) The following defined symbols remain to be analysed: even, COND1, COND2, EVEN, neq, div2, NEQ, DIV2, cond1, cond2 They will be analysed ascendingly in the following order: COND1 = COND2 even < COND1 EVEN < COND1 neq < COND2 div2 < COND2 NEQ < COND2 DIV2 < COND2 even < cond1 neq < cond2 div2 < cond2 cond1 = cond2 ---------------------------------------- (163) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: even(gen_0':s:y10_17(*(2, n15_17))) -> true, rt in Omega(0) Induction Base: even(gen_0':s:y10_17(*(2, 0))) ->_R^Omega(0) true Induction Step: even(gen_0':s:y10_17(*(2, +(n15_17, 1)))) ->_R^Omega(0) even(gen_0':s:y10_17(*(2, n15_17))) ->_IH true We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (164) Obligation: Innermost TRS: Rules: COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0)) COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0')) COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0)) COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0')) COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0)) NEQ(0', 0') -> c5 NEQ(0', s(z0)) -> c6 NEQ(s(z0), 0') -> c7 NEQ(s(z0), s(y)) -> c8(NEQ(z0, y)) EVEN(0') -> c9 EVEN(s(0')) -> c10 EVEN(s(s(z0))) -> c11(EVEN(z0)) DIV2(0') -> c12 DIV2(s(0')) -> c13 DIV2(s(s(z0))) -> c14(DIV2(z0)) P(0') -> c15 P(s(z0)) -> c16 cond1(true, z0) -> cond2(even(z0), z0) cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0)) cond2(false, z0) -> cond1(neq(z0, 0'), p(z0)) neq(0', 0') -> false neq(0', s(z0)) -> true neq(s(z0), 0') -> true neq(s(z0), s(y)) -> neq(z0, y) even(0') -> true even(s(0')) -> false even(s(s(z0))) -> even(z0) div2(0') -> 0' div2(s(0')) -> 0' div2(s(s(z0))) -> s(div2(z0)) p(0') -> 0' p(s(z0)) -> z0 Types: COND1 :: true:false -> 0':s:y -> c true :: true:false c :: c1:c2:c3:c4 -> c9:c10:c11 -> c COND2 :: true:false -> 0':s:y -> c1:c2:c3:c4 even :: 0':s:y -> true:false EVEN :: 0':s:y -> c9:c10:c11 c1 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4 neq :: 0':s:y -> 0':s:y -> true:false 0' :: 0':s:y div2 :: 0':s:y -> 0':s:y NEQ :: 0':s:y -> 0':s:y -> c5:c6:c7:c8 c2 :: c -> c12:c13:c14 -> c1:c2:c3:c4 DIV2 :: 0':s:y -> c12:c13:c14 false :: true:false c3 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4 p :: 0':s:y -> 0':s:y c4 :: c -> c15:c16 -> c1:c2:c3:c4 P :: 0':s:y -> c15:c16 c5 :: c5:c6:c7:c8 s :: 0':s:y -> 0':s:y c6 :: c5:c6:c7:c8 c7 :: c5:c6:c7:c8 y :: 0':s:y c8 :: c5:c6:c7:c8 -> c5:c6:c7:c8 c9 :: c9:c10:c11 c10 :: c9:c10:c11 c11 :: c9:c10:c11 -> c9:c10:c11 c12 :: c12:c13:c14 c13 :: c12:c13:c14 c14 :: c12:c13:c14 -> c12:c13:c14 c15 :: c15:c16 c16 :: c15:c16 cond1 :: true:false -> 0':s:y -> cond1:cond2 cond2 :: true:false -> 0':s:y -> cond1:cond2 hole_c1_17 :: c hole_true:false2_17 :: true:false hole_0':s:y3_17 :: 0':s:y hole_c1:c2:c3:c44_17 :: c1:c2:c3:c4 hole_c9:c10:c115_17 :: c9:c10:c11 hole_c5:c6:c7:c86_17 :: c5:c6:c7:c8 hole_c12:c13:c147_17 :: c12:c13:c14 hole_c15:c168_17 :: c15:c16 hole_cond1:cond29_17 :: cond1:cond2 gen_0':s:y10_17 :: Nat -> 0':s:y gen_c9:c10:c1111_17 :: Nat -> c9:c10:c11 gen_c5:c6:c7:c812_17 :: Nat -> c5:c6:c7:c8 gen_c12:c13:c1413_17 :: Nat -> c12:c13:c14 Lemmas: even(gen_0':s:y10_17(*(2, n15_17))) -> true, rt in Omega(0) Generator Equations: gen_0':s:y10_17(0) <=> 0' gen_0':s:y10_17(+(x, 1)) <=> s(gen_0':s:y10_17(x)) gen_c9:c10:c1111_17(0) <=> c9 gen_c9:c10:c1111_17(+(x, 1)) <=> c11(gen_c9:c10:c1111_17(x)) gen_c5:c6:c7:c812_17(0) <=> c5 gen_c5:c6:c7:c812_17(+(x, 1)) <=> c8(gen_c5:c6:c7:c812_17(x)) gen_c12:c13:c1413_17(0) <=> c12 gen_c12:c13:c1413_17(+(x, 1)) <=> c14(gen_c12:c13:c1413_17(x)) The following defined symbols remain to be analysed: EVEN, COND1, COND2, neq, div2, NEQ, DIV2, cond1, cond2 They will be analysed ascendingly in the following order: COND1 = COND2 EVEN < COND1 neq < COND2 div2 < COND2 NEQ < COND2 DIV2 < COND2 neq < cond2 div2 < cond2 cond1 = cond2 ---------------------------------------- (165) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: EVEN(gen_0':s:y10_17(*(2, n243_17))) -> gen_c9:c10:c1111_17(n243_17), rt in Omega(1 + n243_17) Induction Base: EVEN(gen_0':s:y10_17(*(2, 0))) ->_R^Omega(1) c9 Induction Step: EVEN(gen_0':s:y10_17(*(2, +(n243_17, 1)))) ->_R^Omega(1) c11(EVEN(gen_0':s:y10_17(*(2, n243_17)))) ->_IH c11(gen_c9:c10:c1111_17(c244_17)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (166) Complex Obligation (BEST) ---------------------------------------- (167) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0)) COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0')) COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0)) COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0')) COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0)) NEQ(0', 0') -> c5 NEQ(0', s(z0)) -> c6 NEQ(s(z0), 0') -> c7 NEQ(s(z0), s(y)) -> c8(NEQ(z0, y)) EVEN(0') -> c9 EVEN(s(0')) -> c10 EVEN(s(s(z0))) -> c11(EVEN(z0)) DIV2(0') -> c12 DIV2(s(0')) -> c13 DIV2(s(s(z0))) -> c14(DIV2(z0)) P(0') -> c15 P(s(z0)) -> c16 cond1(true, z0) -> cond2(even(z0), z0) cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0)) cond2(false, z0) -> cond1(neq(z0, 0'), p(z0)) neq(0', 0') -> false neq(0', s(z0)) -> true neq(s(z0), 0') -> true neq(s(z0), s(y)) -> neq(z0, y) even(0') -> true even(s(0')) -> false even(s(s(z0))) -> even(z0) div2(0') -> 0' div2(s(0')) -> 0' div2(s(s(z0))) -> s(div2(z0)) p(0') -> 0' p(s(z0)) -> z0 Types: COND1 :: true:false -> 0':s:y -> c true :: true:false c :: c1:c2:c3:c4 -> c9:c10:c11 -> c COND2 :: true:false -> 0':s:y -> c1:c2:c3:c4 even :: 0':s:y -> true:false EVEN :: 0':s:y -> c9:c10:c11 c1 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4 neq :: 0':s:y -> 0':s:y -> true:false 0' :: 0':s:y div2 :: 0':s:y -> 0':s:y NEQ :: 0':s:y -> 0':s:y -> c5:c6:c7:c8 c2 :: c -> c12:c13:c14 -> c1:c2:c3:c4 DIV2 :: 0':s:y -> c12:c13:c14 false :: true:false c3 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4 p :: 0':s:y -> 0':s:y c4 :: c -> c15:c16 -> c1:c2:c3:c4 P :: 0':s:y -> c15:c16 c5 :: c5:c6:c7:c8 s :: 0':s:y -> 0':s:y c6 :: c5:c6:c7:c8 c7 :: c5:c6:c7:c8 y :: 0':s:y c8 :: c5:c6:c7:c8 -> c5:c6:c7:c8 c9 :: c9:c10:c11 c10 :: c9:c10:c11 c11 :: c9:c10:c11 -> c9:c10:c11 c12 :: c12:c13:c14 c13 :: c12:c13:c14 c14 :: c12:c13:c14 -> c12:c13:c14 c15 :: c15:c16 c16 :: c15:c16 cond1 :: true:false -> 0':s:y -> cond1:cond2 cond2 :: true:false -> 0':s:y -> cond1:cond2 hole_c1_17 :: c hole_true:false2_17 :: true:false hole_0':s:y3_17 :: 0':s:y hole_c1:c2:c3:c44_17 :: c1:c2:c3:c4 hole_c9:c10:c115_17 :: c9:c10:c11 hole_c5:c6:c7:c86_17 :: c5:c6:c7:c8 hole_c12:c13:c147_17 :: c12:c13:c14 hole_c15:c168_17 :: c15:c16 hole_cond1:cond29_17 :: cond1:cond2 gen_0':s:y10_17 :: Nat -> 0':s:y gen_c9:c10:c1111_17 :: Nat -> c9:c10:c11 gen_c5:c6:c7:c812_17 :: Nat -> c5:c6:c7:c8 gen_c12:c13:c1413_17 :: Nat -> c12:c13:c14 Lemmas: even(gen_0':s:y10_17(*(2, n15_17))) -> true, rt in Omega(0) Generator Equations: gen_0':s:y10_17(0) <=> 0' gen_0':s:y10_17(+(x, 1)) <=> s(gen_0':s:y10_17(x)) gen_c9:c10:c1111_17(0) <=> c9 gen_c9:c10:c1111_17(+(x, 1)) <=> c11(gen_c9:c10:c1111_17(x)) gen_c5:c6:c7:c812_17(0) <=> c5 gen_c5:c6:c7:c812_17(+(x, 1)) <=> c8(gen_c5:c6:c7:c812_17(x)) gen_c12:c13:c1413_17(0) <=> c12 gen_c12:c13:c1413_17(+(x, 1)) <=> c14(gen_c12:c13:c1413_17(x)) The following defined symbols remain to be analysed: EVEN, COND1, COND2, neq, div2, NEQ, DIV2, cond1, cond2 They will be analysed ascendingly in the following order: COND1 = COND2 EVEN < COND1 neq < COND2 div2 < COND2 NEQ < COND2 DIV2 < COND2 neq < cond2 div2 < cond2 cond1 = cond2 ---------------------------------------- (168) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (169) BOUNDS(n^1, INF) ---------------------------------------- (170) Obligation: Innermost TRS: Rules: COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0)) COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0')) COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0)) COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0')) COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0)) NEQ(0', 0') -> c5 NEQ(0', s(z0)) -> c6 NEQ(s(z0), 0') -> c7 NEQ(s(z0), s(y)) -> c8(NEQ(z0, y)) EVEN(0') -> c9 EVEN(s(0')) -> c10 EVEN(s(s(z0))) -> c11(EVEN(z0)) DIV2(0') -> c12 DIV2(s(0')) -> c13 DIV2(s(s(z0))) -> c14(DIV2(z0)) P(0') -> c15 P(s(z0)) -> c16 cond1(true, z0) -> cond2(even(z0), z0) cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0)) cond2(false, z0) -> cond1(neq(z0, 0'), p(z0)) neq(0', 0') -> false neq(0', s(z0)) -> true neq(s(z0), 0') -> true neq(s(z0), s(y)) -> neq(z0, y) even(0') -> true even(s(0')) -> false even(s(s(z0))) -> even(z0) div2(0') -> 0' div2(s(0')) -> 0' div2(s(s(z0))) -> s(div2(z0)) p(0') -> 0' p(s(z0)) -> z0 Types: COND1 :: true:false -> 0':s:y -> c true :: true:false c :: c1:c2:c3:c4 -> c9:c10:c11 -> c COND2 :: true:false -> 0':s:y -> c1:c2:c3:c4 even :: 0':s:y -> true:false EVEN :: 0':s:y -> c9:c10:c11 c1 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4 neq :: 0':s:y -> 0':s:y -> true:false 0' :: 0':s:y div2 :: 0':s:y -> 0':s:y NEQ :: 0':s:y -> 0':s:y -> c5:c6:c7:c8 c2 :: c -> c12:c13:c14 -> c1:c2:c3:c4 DIV2 :: 0':s:y -> c12:c13:c14 false :: true:false c3 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4 p :: 0':s:y -> 0':s:y c4 :: c -> c15:c16 -> c1:c2:c3:c4 P :: 0':s:y -> c15:c16 c5 :: c5:c6:c7:c8 s :: 0':s:y -> 0':s:y c6 :: c5:c6:c7:c8 c7 :: c5:c6:c7:c8 y :: 0':s:y c8 :: c5:c6:c7:c8 -> c5:c6:c7:c8 c9 :: c9:c10:c11 c10 :: c9:c10:c11 c11 :: c9:c10:c11 -> c9:c10:c11 c12 :: c12:c13:c14 c13 :: c12:c13:c14 c14 :: c12:c13:c14 -> c12:c13:c14 c15 :: c15:c16 c16 :: c15:c16 cond1 :: true:false -> 0':s:y -> cond1:cond2 cond2 :: true:false -> 0':s:y -> cond1:cond2 hole_c1_17 :: c hole_true:false2_17 :: true:false hole_0':s:y3_17 :: 0':s:y hole_c1:c2:c3:c44_17 :: c1:c2:c3:c4 hole_c9:c10:c115_17 :: c9:c10:c11 hole_c5:c6:c7:c86_17 :: c5:c6:c7:c8 hole_c12:c13:c147_17 :: c12:c13:c14 hole_c15:c168_17 :: c15:c16 hole_cond1:cond29_17 :: cond1:cond2 gen_0':s:y10_17 :: Nat -> 0':s:y gen_c9:c10:c1111_17 :: Nat -> c9:c10:c11 gen_c5:c6:c7:c812_17 :: Nat -> c5:c6:c7:c8 gen_c12:c13:c1413_17 :: Nat -> c12:c13:c14 Lemmas: even(gen_0':s:y10_17(*(2, n15_17))) -> true, rt in Omega(0) EVEN(gen_0':s:y10_17(*(2, n243_17))) -> gen_c9:c10:c1111_17(n243_17), rt in Omega(1 + n243_17) Generator Equations: gen_0':s:y10_17(0) <=> 0' gen_0':s:y10_17(+(x, 1)) <=> s(gen_0':s:y10_17(x)) gen_c9:c10:c1111_17(0) <=> c9 gen_c9:c10:c1111_17(+(x, 1)) <=> c11(gen_c9:c10:c1111_17(x)) gen_c5:c6:c7:c812_17(0) <=> c5 gen_c5:c6:c7:c812_17(+(x, 1)) <=> c8(gen_c5:c6:c7:c812_17(x)) gen_c12:c13:c1413_17(0) <=> c12 gen_c12:c13:c1413_17(+(x, 1)) <=> c14(gen_c12:c13:c1413_17(x)) The following defined symbols remain to be analysed: neq, COND1, COND2, div2, NEQ, DIV2, cond1, cond2 They will be analysed ascendingly in the following order: COND1 = COND2 neq < COND2 div2 < COND2 NEQ < COND2 DIV2 < COND2 neq < cond2 div2 < cond2 cond1 = cond2 ---------------------------------------- (171) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: div2(gen_0':s:y10_17(*(2, n779_17))) -> gen_0':s:y10_17(n779_17), rt in Omega(0) Induction Base: div2(gen_0':s:y10_17(*(2, 0))) ->_R^Omega(0) 0' Induction Step: div2(gen_0':s:y10_17(*(2, +(n779_17, 1)))) ->_R^Omega(0) s(div2(gen_0':s:y10_17(*(2, n779_17)))) ->_IH s(gen_0':s:y10_17(c780_17)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (172) Obligation: Innermost TRS: Rules: COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0)) COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0')) COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0)) COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0')) COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0)) NEQ(0', 0') -> c5 NEQ(0', s(z0)) -> c6 NEQ(s(z0), 0') -> c7 NEQ(s(z0), s(y)) -> c8(NEQ(z0, y)) EVEN(0') -> c9 EVEN(s(0')) -> c10 EVEN(s(s(z0))) -> c11(EVEN(z0)) DIV2(0') -> c12 DIV2(s(0')) -> c13 DIV2(s(s(z0))) -> c14(DIV2(z0)) P(0') -> c15 P(s(z0)) -> c16 cond1(true, z0) -> cond2(even(z0), z0) cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0)) cond2(false, z0) -> cond1(neq(z0, 0'), p(z0)) neq(0', 0') -> false neq(0', s(z0)) -> true neq(s(z0), 0') -> true neq(s(z0), s(y)) -> neq(z0, y) even(0') -> true even(s(0')) -> false even(s(s(z0))) -> even(z0) div2(0') -> 0' div2(s(0')) -> 0' div2(s(s(z0))) -> s(div2(z0)) p(0') -> 0' p(s(z0)) -> z0 Types: COND1 :: true:false -> 0':s:y -> c true :: true:false c :: c1:c2:c3:c4 -> c9:c10:c11 -> c COND2 :: true:false -> 0':s:y -> c1:c2:c3:c4 even :: 0':s:y -> true:false EVEN :: 0':s:y -> c9:c10:c11 c1 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4 neq :: 0':s:y -> 0':s:y -> true:false 0' :: 0':s:y div2 :: 0':s:y -> 0':s:y NEQ :: 0':s:y -> 0':s:y -> c5:c6:c7:c8 c2 :: c -> c12:c13:c14 -> c1:c2:c3:c4 DIV2 :: 0':s:y -> c12:c13:c14 false :: true:false c3 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4 p :: 0':s:y -> 0':s:y c4 :: c -> c15:c16 -> c1:c2:c3:c4 P :: 0':s:y -> c15:c16 c5 :: c5:c6:c7:c8 s :: 0':s:y -> 0':s:y c6 :: c5:c6:c7:c8 c7 :: c5:c6:c7:c8 y :: 0':s:y c8 :: c5:c6:c7:c8 -> c5:c6:c7:c8 c9 :: c9:c10:c11 c10 :: c9:c10:c11 c11 :: c9:c10:c11 -> c9:c10:c11 c12 :: c12:c13:c14 c13 :: c12:c13:c14 c14 :: c12:c13:c14 -> c12:c13:c14 c15 :: c15:c16 c16 :: c15:c16 cond1 :: true:false -> 0':s:y -> cond1:cond2 cond2 :: true:false -> 0':s:y -> cond1:cond2 hole_c1_17 :: c hole_true:false2_17 :: true:false hole_0':s:y3_17 :: 0':s:y hole_c1:c2:c3:c44_17 :: c1:c2:c3:c4 hole_c9:c10:c115_17 :: c9:c10:c11 hole_c5:c6:c7:c86_17 :: c5:c6:c7:c8 hole_c12:c13:c147_17 :: c12:c13:c14 hole_c15:c168_17 :: c15:c16 hole_cond1:cond29_17 :: cond1:cond2 gen_0':s:y10_17 :: Nat -> 0':s:y gen_c9:c10:c1111_17 :: Nat -> c9:c10:c11 gen_c5:c6:c7:c812_17 :: Nat -> c5:c6:c7:c8 gen_c12:c13:c1413_17 :: Nat -> c12:c13:c14 Lemmas: even(gen_0':s:y10_17(*(2, n15_17))) -> true, rt in Omega(0) EVEN(gen_0':s:y10_17(*(2, n243_17))) -> gen_c9:c10:c1111_17(n243_17), rt in Omega(1 + n243_17) div2(gen_0':s:y10_17(*(2, n779_17))) -> gen_0':s:y10_17(n779_17), rt in Omega(0) Generator Equations: gen_0':s:y10_17(0) <=> 0' gen_0':s:y10_17(+(x, 1)) <=> s(gen_0':s:y10_17(x)) gen_c9:c10:c1111_17(0) <=> c9 gen_c9:c10:c1111_17(+(x, 1)) <=> c11(gen_c9:c10:c1111_17(x)) gen_c5:c6:c7:c812_17(0) <=> c5 gen_c5:c6:c7:c812_17(+(x, 1)) <=> c8(gen_c5:c6:c7:c812_17(x)) gen_c12:c13:c1413_17(0) <=> c12 gen_c12:c13:c1413_17(+(x, 1)) <=> c14(gen_c12:c13:c1413_17(x)) The following defined symbols remain to be analysed: NEQ, COND1, COND2, DIV2, cond1, cond2 They will be analysed ascendingly in the following order: COND1 = COND2 NEQ < COND2 DIV2 < COND2 cond1 = cond2 ---------------------------------------- (173) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: DIV2(gen_0':s:y10_17(*(2, n1244_17))) -> gen_c12:c13:c1413_17(n1244_17), rt in Omega(1 + n1244_17) Induction Base: DIV2(gen_0':s:y10_17(*(2, 0))) ->_R^Omega(1) c12 Induction Step: DIV2(gen_0':s:y10_17(*(2, +(n1244_17, 1)))) ->_R^Omega(1) c14(DIV2(gen_0':s:y10_17(*(2, n1244_17)))) ->_IH c14(gen_c12:c13:c1413_17(c1245_17)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (174) Obligation: Innermost TRS: Rules: COND1(true, z0) -> c(COND2(even(z0), z0), EVEN(z0)) COND2(true, z0) -> c1(COND1(neq(z0, 0'), div2(z0)), NEQ(z0, 0')) COND2(true, z0) -> c2(COND1(neq(z0, 0'), div2(z0)), DIV2(z0)) COND2(false, z0) -> c3(COND1(neq(z0, 0'), p(z0)), NEQ(z0, 0')) COND2(false, z0) -> c4(COND1(neq(z0, 0'), p(z0)), P(z0)) NEQ(0', 0') -> c5 NEQ(0', s(z0)) -> c6 NEQ(s(z0), 0') -> c7 NEQ(s(z0), s(y)) -> c8(NEQ(z0, y)) EVEN(0') -> c9 EVEN(s(0')) -> c10 EVEN(s(s(z0))) -> c11(EVEN(z0)) DIV2(0') -> c12 DIV2(s(0')) -> c13 DIV2(s(s(z0))) -> c14(DIV2(z0)) P(0') -> c15 P(s(z0)) -> c16 cond1(true, z0) -> cond2(even(z0), z0) cond2(true, z0) -> cond1(neq(z0, 0'), div2(z0)) cond2(false, z0) -> cond1(neq(z0, 0'), p(z0)) neq(0', 0') -> false neq(0', s(z0)) -> true neq(s(z0), 0') -> true neq(s(z0), s(y)) -> neq(z0, y) even(0') -> true even(s(0')) -> false even(s(s(z0))) -> even(z0) div2(0') -> 0' div2(s(0')) -> 0' div2(s(s(z0))) -> s(div2(z0)) p(0') -> 0' p(s(z0)) -> z0 Types: COND1 :: true:false -> 0':s:y -> c true :: true:false c :: c1:c2:c3:c4 -> c9:c10:c11 -> c COND2 :: true:false -> 0':s:y -> c1:c2:c3:c4 even :: 0':s:y -> true:false EVEN :: 0':s:y -> c9:c10:c11 c1 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4 neq :: 0':s:y -> 0':s:y -> true:false 0' :: 0':s:y div2 :: 0':s:y -> 0':s:y NEQ :: 0':s:y -> 0':s:y -> c5:c6:c7:c8 c2 :: c -> c12:c13:c14 -> c1:c2:c3:c4 DIV2 :: 0':s:y -> c12:c13:c14 false :: true:false c3 :: c -> c5:c6:c7:c8 -> c1:c2:c3:c4 p :: 0':s:y -> 0':s:y c4 :: c -> c15:c16 -> c1:c2:c3:c4 P :: 0':s:y -> c15:c16 c5 :: c5:c6:c7:c8 s :: 0':s:y -> 0':s:y c6 :: c5:c6:c7:c8 c7 :: c5:c6:c7:c8 y :: 0':s:y c8 :: c5:c6:c7:c8 -> c5:c6:c7:c8 c9 :: c9:c10:c11 c10 :: c9:c10:c11 c11 :: c9:c10:c11 -> c9:c10:c11 c12 :: c12:c13:c14 c13 :: c12:c13:c14 c14 :: c12:c13:c14 -> c12:c13:c14 c15 :: c15:c16 c16 :: c15:c16 cond1 :: true:false -> 0':s:y -> cond1:cond2 cond2 :: true:false -> 0':s:y -> cond1:cond2 hole_c1_17 :: c hole_true:false2_17 :: true:false hole_0':s:y3_17 :: 0':s:y hole_c1:c2:c3:c44_17 :: c1:c2:c3:c4 hole_c9:c10:c115_17 :: c9:c10:c11 hole_c5:c6:c7:c86_17 :: c5:c6:c7:c8 hole_c12:c13:c147_17 :: c12:c13:c14 hole_c15:c168_17 :: c15:c16 hole_cond1:cond29_17 :: cond1:cond2 gen_0':s:y10_17 :: Nat -> 0':s:y gen_c9:c10:c1111_17 :: Nat -> c9:c10:c11 gen_c5:c6:c7:c812_17 :: Nat -> c5:c6:c7:c8 gen_c12:c13:c1413_17 :: Nat -> c12:c13:c14 Lemmas: even(gen_0':s:y10_17(*(2, n15_17))) -> true, rt in Omega(0) EVEN(gen_0':s:y10_17(*(2, n243_17))) -> gen_c9:c10:c1111_17(n243_17), rt in Omega(1 + n243_17) div2(gen_0':s:y10_17(*(2, n779_17))) -> gen_0':s:y10_17(n779_17), rt in Omega(0) DIV2(gen_0':s:y10_17(*(2, n1244_17))) -> gen_c12:c13:c1413_17(n1244_17), rt in Omega(1 + n1244_17) Generator Equations: gen_0':s:y10_17(0) <=> 0' gen_0':s:y10_17(+(x, 1)) <=> s(gen_0':s:y10_17(x)) gen_c9:c10:c1111_17(0) <=> c9 gen_c9:c10:c1111_17(+(x, 1)) <=> c11(gen_c9:c10:c1111_17(x)) gen_c5:c6:c7:c812_17(0) <=> c5 gen_c5:c6:c7:c812_17(+(x, 1)) <=> c8(gen_c5:c6:c7:c812_17(x)) gen_c12:c13:c1413_17(0) <=> c12 gen_c12:c13:c1413_17(+(x, 1)) <=> c14(gen_c12:c13:c1413_17(x)) The following defined symbols remain to be analysed: cond2, COND1, COND2, cond1 They will be analysed ascendingly in the following order: COND1 = COND2 cond1 = cond2